Chapter 5 — Arithmetic Progressions

Sequence: A sequence is an ordered list of numbers written as a1, a2, a3, ... Each number is called a term. A sequence can be finite (has a last term) or infinite.

Series: A series is the sum of terms of a sequence. For example, a1 + a2 + a3 + ... is a series.

Arithmetic Progression (AP): An AP is a sequence in which the difference between any two consecutive terms is constant. This constant is called the common difference and is denoted by d. If the first term is a, the AP is: a, a + d, a + 2d, a + 3d, ...

nth term of an AP (derivation): The first term is a (n = 1). The second is a + d (n = 2). Continuing, the nth term a_n equals a + (n − 1)d. So a_n = a + (n − 1)d. This gives a direct formula to find any term without listing all previous terms.

Arithmetic Series (sum of an AP): The sum of first n terms of an AP is denoted by S_n. Using the pairing method (first + last, second + second-last, ...), we get the formula:

S_n = n/2 [2a + (n − 1)d]

Equivalently, if l is the nth (last) term, S_n = n/2 (a + l).

Remarks:

• If d = 0, the AP is constant: all terms equal a.
• If d > 0 the AP is increasing; if d < 0 it is decreasing.
• An AP plotted as points (index n on x-axis, term a_n on y-axis) lies on a straight line; the slope equals d.

Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference and is denoted by d.

If the first term of an AP is a₁ (often written simply as a) and the common difference is d, the sequence looks like:

a, a + d, a + 2d, a + 3d, ...

The general (nth) term of an AP is given by:

a_n = a₁ + (n − 1)·d

Key points:

• Common difference d = a_k+1 − a_k for any valid k. If d = 0, all terms are equal.
• An AP can be finite (has a last term) or infinite.
• To check whether a sequence is an AP, compute consecutive differences; if all equal, it is an AP.
• For two terms a_m and a_n, a_n = a_m + (n − m)·d.

What is an AP? An arithmetic progression (AP) is a sequence of numbers in which the difference between any two successive terms is constant. This constant is called the common difference (d).

Definition of the nth term (general term): If a is the first term and d is the common difference of an AP, then the nth term (denoted by a_n) is

a_n = a + (n - 1)d

Derivation (brief): Terms: a, a + d, a + 2d, ... The 1st term is a, the 2nd term is a + d, so the nth term increases d each step from the first: a_n = a + (n - 1)d.

Other useful forms:

• If you know two terms a_m and a_n (m & n are their positions), the common difference is d = (a_n - a_m) / (n - m). Then the general term can be written as a_k = a_m + (k - m)d for any k.
• If l is the last (N-th) term, then a_N = l = a + (N - 1)d, and the nth term can also be written as a_n = l - (N - n)d.
• To find the position n of a given term T: n = ((T - a) / d) + 1 (valid when d ≠ 0 and (T - a) is divisible by d in integer sequences).

Important remarks:

• n is a positive integer (1, 2, 3, ...).
• When d > 0 the sequence is increasing; when d < 0 it is decreasing; when d = 0 all terms are equal.

What is an AP? An arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant is called the common difference and is denoted by d. If a is the first term, the sequence is: a, a + d, a + 2d, ...

Last term (general idea)
The last term of a finite AP with n terms is the n-th term. Using the standard nth-term formula, the last term (often denoted by l or a_n) is given by:

a_n = a + (n - 1)d

Derivation (brief)
Start from the sequence: a (1st), a + d (2nd), a + 2d (3rd), ... For the n-th term you add the common difference (n-1) times to the first term, so a_n = a + (n-1)d. For a finite AP with n terms, this a_n is the last term.

Using the sum to find the last term
If the sum S_n of the first n terms is known, the last term can be obtained from the sum formula:

S_n = n/2 (a + l) => l = (2 S_n)/n - a

Other useful forms
If the first term a, last term l and number of terms n are known, the common difference can be found: d = (l - a)/(n - 1). Special cases: if d = 0 the AP is constant and the last term equals the first term; if d > 0 the sequence increases and if d < 0 it decreases.

Important notes
- "Last term" only makes sense for a finite AP (one with a specified number of terms n).
- n must be a positive integer. If given a potential last term value, check whether it fits integer-index condition: (l - a) must be a multiple of d and (l - a)/d must equal n - 1 (an integer ≥ 0).

An arithmetic progression (AP) is a sequence in which each term after the first is obtained by adding a constant difference (d) to the previous term. If a is the first term, the terms are: a, a + d, a + 2d, ..., a + (n-1)d.

We often need the sum of the first n terms, denoted S_n. There are two common and useful forms of the formula for S_n. One uses the last term (l) and the other uses the common difference (d).

1. Derivation by pairing:

   Write the sum forwards and backwards and add term-wise:

   S_n = a + (a+d) + (a+2d) + ... + [a+(n-1)d]

   Also, S_n = [a+(n-1)d] + [a+(n-2)d] + ... + a

   Add both expressions term by term. Each pair (first + last), (second + second last), ... equals 2a + (n-1)d. There are n such pairs, so 2S_n = n[2a + (n-1)d].

   Therefore S_n = (n/2) [2a + (n-1)d].

2. Using the last term:

   If the nth term is l = a + (n-1)d, then the sum can be written as S_n = (n/2)(a + l). This follows because each pair adds to (a + l).

3. Remarks and special cases:
   • If d = 0 (all terms equal a), S_n = n·a.
   • For the sequence of natural numbers 1, 2, 3, ..., we have a = 1, d = 1, so S_n = n(n+1)/2.
   • If d is negative, the same formula applies (sum may decrease as n grows).

These formulas let you compute sums quickly without adding every term individually.

Definition: An arithmetic progression (AP) is a sequence of numbers in which the difference between any two successive terms is constant. If a is the first term and d is the common difference, the nth term is a_n = a + (n-1)d.

Basic Observations:

• The common difference d = a_{n+1} - a_n is constant for all n.
• If d > 0 the AP is increasing; if d < 0 it is decreasing; if d = 0 all terms are equal.
• Linear transformation: adding a constant to every term or multiplying every term by a fixed nonzero constant results in another AP (with adjusted a and d).

Key Properties and Useful Results:

• nth term formula: a_n = a + (n-1)d.
• Sum of first n terms: S_n = n/2(2a + (n-1)d) = n(a_1 + a_n)/2. This is often written as S_n = n/2( a + l ), where l = a_n is the last term.
• Equidistant (symmetric) terms: In an AP, terms equidistant from the beginning and end have the same sum: a_k + a_{n-k+1} = a_1 + a_n (for 1 ≤ k ≤ n). This is why S_n can be computed by pairing first and last terms.
• Arithmetic mean (three-term condition): Three numbers p, q, r are in AP iff 2q = p + r. More generally, any term of an AP equals the average of the two terms equally distant from it: a_n = (a_{n-k} + a_{n+k})/2.
• Finding d from two terms: If a_m = x and a_n = y (m < n), then d = (y - x)/(n - m) and a = x - (m-1)d.
• Number of terms when last term known: If last term l is given, n = ((l - a)/d) + 1 (provided d ≠ 0 and the result is a positive integer).
• Inserting k arithmetic means: To insert k arithmetic means between numbers A and B, treat the whole sequence as an AP with (k+2) terms. Common difference d = (B - A)/(k+1). The inserted means are A + d, A + 2d, …, A + kd.
• Consecutive-difference property: Differences of consecutive terms are constant; differences of order higher than 1 are zero (i.e., second differences = 0). This is why plotting a_n vs n gives a straight line.

Tips for Problem Solving: Use the nth term formula to translate between indices and values; use the symmetric-sum property to compute sums quickly by pairing; use the arithmetic-mean condition when checking whether numbers are in AP or when inserting means; express unknowns (a and d) and solve linear equations when two terms or a term and sum are given.

Definition (Arithmetic Mean, AM): For two numbers a and b, the arithmetic mean is the number (a + b) / 2. More generally, numbers that form an arithmetic progression (AP) have constant differences; any term that lies exactly midway between two terms of an AP is their arithmetic mean.

Insertion of Arithmetic Means: Given two numbers a and b, to insert n arithmetic means between them means to find n numbers so that the sequence a, (n means...), b forms an AP. In other words the entire list of n+2 terms must have a common difference d.

Derivation and key result:

• Let the AP have first term a and last term b, and let n terms be inserted. Total number of terms = n + 2.
• Common difference d = (b - a) / (n + 1). This ensures equal spacing from a to b in (n+1) intervals.
• The k-th inserted mean (k = 1,2,...,n) is: a + k·d = a + k·(b - a)/(n + 1).

Useful properties:

• For three numbers x, y, z to be in AP, y = (x + z) / 2.
• Sum of the n inserted arithmetic means = n·(a + b) / 2. (Quick derivation: sum of inserted = n·a + [n(n+1)/2]·d with d=(b-a)/(n+1) simplifies to n·(a+b)/2.)
• If terms of an AP are written as a1, a2, ..., then general term: a_r = a1 + (r - 1)·d.

How to solve insertion problems (step-by-step):

1. Identify a (first term), b (last term) and n (number of means to insert).
2. Compute d = (b - a) / (n + 1).
3. Write the inserted terms as a + d, a + 2d, ..., a + n·d.
4. Optionally, check by verifying each consecutive difference equals d and the final term equals b.

Why this matters (intuition): Inserting arithmetic means is equivalent to dividing the interval from a to b into (n + 1) equal steps. On a number line or on a straight line graph of index vs value, these inserted points are evenly spaced.

What is an arithmetic progression (AP)?
An AP is a sequence of numbers in which each term after the first is obtained by adding a fixed number called the common difference (d). If the first term is a, the nth term is written as a_n.

Key relations

• nth term: a_n = a + (n - 1)d
• sum of first n terms: S_n = (n/2) [2a + (n - 1)d]

How to determine the common difference (d)

• From two known terms: if you know a_m and a_n (m <> n), then
  d = (a_m - a_n) / (m - n).
• From first term, last term and number of terms: if last term (l) is known, then
  d = (l - a) / (n - 1).

How to determine the number of terms (n)

• When first term a, common difference d, and last term l are known:
  use a_n = l = a + (n - 1)d → n = ((l - a) / d) + 1. (n must be a positive integer.)
• When first term a, common difference d, and sum S_n are known:
  use S_n = (n/2)[2a + (n - 1)d]. This gives a quadratic in n:
  n(2a + (n - 1)d) = 2S_n → rearrange to standard quadratic and solve with the quadratic formula.
  Keep only the positive integer solution(s) because n must be a positive integer.

Practical tips and checks

• Always check that the computed n is a positive integer; if not, re-check given data or sign of d.
• The sign of d indicates direction: d > 0 means increasing sequence, d < 0 means decreasing.
• When solving the quadratic for n, discard negative and non-integer roots.

Arithmetic Progression (AP) problems and applications focus on sequences where each term after the first is obtained by adding a fixed number (common difference) to the previous term. The chapter "Problem Types and Applications" trains students to recognize AP patterns, form equations, and solve practical problems using AP formulas.

• Basic identification: Recognise an AP from a list of terms and find the first term a and common difference d.
• Find n-th term: Use the nth-term formula to compute any term.
• Sum of terms: Compute the sum of first n terms; use different forms of the sum formula depending on whether the last term is known.
• Reverse problems: Given two terms, or a term and a sum, determine a, d, or n by setting up linear equations.
• Word problems / applications: Model real-life situations (seating arrangements, savings with equal increase, arithmetic mean tasks, arrangement of objects) as APs, translate text to equations, then solve.

Strategy for solving AP problems:

1. Translate the words to symbols: denote first term by a, common difference by d, term number by n, nth term by a_n, sum of first n terms by S_n.
2. Write down the appropriate formula(s) (nth term, sum) that relate the given quantities.
3. Substitute given values and solve the resulting algebraic equation(s) for the unknown(s).
4. Check the answer fits the context (integer n, positive seats/money etc.).

Common problem patterns:

• Find a and d when two non-consecutive terms are given: form two linear equations using the nth-term formula.
• Find n when a term or sum is given: rearrange the nth-term or sum formula to solve for n (watch for arithmetic progressions leading to quadratic equations in n when using the sum formula).
• Find sum when only a and d are given: either compute last term a_n first or use the sum formula directly.
• Word problems (rows of chairs, stair steps, installments increasing by fixed amount): translate counts into an AP and sum if total is required.

Understanding these patterns and practicing translation from words to equations makes solving AP applications systematic and reliable.

Definition: An arithmetic progression (AP) is a sequence of numbers in which the difference between any two successive terms is constant. If a is the first term and d is the common difference, the nth term is a_n = a + (n - 1)d.

Graphical idea: To graphically represent an AP, take the term number n on the x-axis (domain: positive integers 1, 2, 3, ...) and the corresponding term a_n on the y-axis. Because a_n = d·n + (a - d), the points (n, a_n) lie on a straight line with slope d and y-intercept (value at n = 0) equal to a - d. Thus the graph of the terms of an AP is a set of equally spaced points that fall on a straight line. (Note: the sequence gives discrete points; joining them gives the underlying linear function.)

Interpretation of slope and intercept:

• Slope = d: rise/run between consecutive plotted points equals the common difference.
• y-intercept = a - d: if you extend the line to n = 0, the y-value equals a - d. The first term a is at n = 1, so a = (y-intercept) + d.

How to read an AP from its graph:

• Pick any two plotted points (n1, a_{n1}) and (n2, a_{n2}). The common difference d = (a_{n2} - a_{n1})/(n2 - n1).
• Find a by reading the value at n = 1, or compute a = a_n - (n - 1)d for any plotted point.

Discrete vs continuous view: The sequence is discrete (points at integer n). For visualization or modelling you may draw the straight line that extends those points; that line corresponds to the linear function f(n) = d·n + (a - d).

Overview: An arithmetic progression (AP) is a sequence of numbers in which the difference between consecutive terms is constant. Let the first term be a, common difference d, and the nth term be t_n. The basic formulas are t_n = a + (n-1)d and S_n = n/2 [2a + (n-1)d] (sum of first n terms). This topic covers commonly encountered special cases and helpful properties that simplify working with APs.

Key special cases and properties

1. d = 0 (constant sequence): Every term equals a. So t_n = a for all n and S_n = n·a.
2. d > 0 or d < 0: If d > 0 the AP is increasing; if d < 0 it is decreasing. The plot of t_n versus n is a straight line with slope d.
3. Last term given (l): If the last term l is known, use l = a + (n-1)d to find n or d. Sum becomes S_n = n/2 (a + l).
4. Pairing property: In any AP, terms equidistant from the ends add to the same value: t_k + t_{n-k+1} = a + l = 2·(arithmetic mean). This is the reason S_n = n/2 (a + l) works.
5. Middle term and average: If n is odd, the middle term (t_{(n+1)/2}) equals the arithmetic mean of the sequence: t_{(n+1)/2} = S_n / n = (a + l)/2. For any n, the average of the sequence equals (first + last)/2.
6. a = 0: Terms are 0, d, 2d, ...; then t_n = (n-1)d and S_n = n/2 [(n-1)d] = (d/2) n(n-1), a quadratic in n.
7. Sum as a quadratic function: With d ≠ 0, S_n = n/2 [2a + (n-1)d] is a quadratic polynomial in n. So the graph of S_n vs n is a parabola (discrete points lie on a parabola).
8. When a term or sum becomes zero: To find n for which t_n = 0, solve a + (n-1)d = 0. To solve S_n = 0, solve the quadratic n[2a + (n-1)d] = 0 (gives n = 0 or a relation for n).

Derivation (pairing method) for S_n: Write S_n = a + (a + d) + ... + (a + (n-1)d). Write it forwards and backwards and add pairwise; each pair sums to (2a + (n-1)d). There are n pairs in the combined sum, so 2S_n = n[2a + (n-1)d], hence S_n = n/2 [2a + (n-1)d].

Usefulness in problems: These special cases let you quickly compute terms and sums, check for arithmetic means, solve for n or d when sums/terms are given, and identify when sequences are constant or symmetric about 0. Visualizing t_n as a straight line and S_n as a parabola helps understand growth patterns.

What this topic means
"Connections and Extensions" in the study of Arithmetic Progressions (AP) highlights how AP links to other areas of mathematics and to everyday problems. It shows the algebraic, geometric and graphical viewpoints, and how AP formulas extend to solve practical situations.

Core ideas (brief)

• An AP is a sequence with constant difference d: a, a+d, a+2d, ...
• nth term: a_n = a + (n-1)d. This is a linear function of n (so plotting n vs a_n gives a straight line).
• Sum of first n terms: S_n = n/2(2a + (n-1)d) = n/2(a_1 + a_n). This is a quadratic function of n (so plotting n vs S_n gives a parabola).
• Arithmetic mean and midpoint property: for terms equidistant from ends, their average equals the middle value: (a_k + a_{n-k+1})/2 = (a_1 + a_n)/2.

Connections

• Algebra: a_n is a first-degree expression in n. Solving for n or d reduces to linear equations.
• Coordinate geometry: points (n, a_n) lie on a straight line with slope d; slope-intercept form gives geometric insight into difference and starting term.
• Quadratic sequences and sums: S_n is quadratic, so cumulative totals (money saved, total seats filled) grow like a parabola; this links AP sums to quadratic functions.
• Number line and geometry: equal spacing of terms models equal partitioning and is used in arranging seats, plants, steps etc.
• Problem modelling: salary increments, installment plans, arithmetic packing/arrangement and many counting problems can be formulated as APs.

How to extend and use AP ideas

• Translate real-life data into a_n = a + (n-1)d, then apply formulas for term or sum.
• Use the linear nature to interpolate or extrapolate values (predict future terms).
• Recognize when a sequence is not AP (non-constant differences) and then consider second differences—if constant, underlying rule is quadratic (related to sums of AP).

Important observations

• If you know two terms a_i and a_j, d = (a_j - a_i)/(j-i).
• For odd number of terms, the middle term equals the arithmetic mean of the sequence: a_{(n+1)/2} = S_n/n.
• Summing pairwise terms from ends: a_1 + a_n = a_2 + a_{n-1} = constant; this simplifies many computations.