Introduction
An Arithmetic Progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a fixed number called the common difference to the preceding term. APs appear frequently in real life — from salary increments to seating arrangements. This chapter covers finding the nth term and the sum of the first n terms of an AP.
Introduction to Arithmetic Progressions
An Arithmetic Progression is a list of numbers in which each term is obtained by adding a fixed number 'd' (called the common difference) to the preceding term. The first term is denoted by 'a'. So an AP looks like: a, a+d, a+2d, a+3d, ... The common difference can be positive (increasing AP), negative (decreasing AP), or zero (constant AP).
Key Points
- •An AP has a constant common difference between consecutive terms
- •Common difference d = a2 - a1 = a3 - a2 = ... (any term minus previous term)
- •d can be positive, negative, or zero
- •A sequence is an AP if and only if the difference between consecutive terms is constant
- •First term 'a' and common difference 'd' completely define an AP
Worked Example
Is 2, 5, 8, 11, 14, ... an AP? d = 5 - 2 = 3 d = 8 - 5 = 3 d = 11 - 8 = 3 Since the common difference is constant (d = 3), this is an AP with a = 2, d = 3.
Watch Out
To check if a sequence is an AP, verify that the difference between every pair of consecutive terms is the same.
nth Term of an AP
The nth term (also called the general term) of an AP with first term 'a' and common difference 'd' is given by an = a + (n-1)d. This formula lets us find any term of the AP without listing all previous terms. It can also be used to find the number of terms, or the value of a or d if sufficient information is given.
Key Points
- •nth term: an = a + (n-1)d
- •The nth term from the end = l - (n-1)d, where l is the last term
- •If an = a + (n-1)d is given a specific value, you can find n
- •Three numbers a, b, c are in AP if 2b = a + c
- •The common difference d = (an - a) / (n-1)
Worked Example
Find the 20th term of the AP: 3, 7, 11, 15, ... a = 3, d = 7 - 3 = 4 a20 = a + (20-1)d = 3 + 19(4) = 3 + 76 = 79 The 20th term is 79.
Watch Out
When a question says 'which term of the AP is...', set an equal to the given value and solve for n. If n is a positive integer, it is a term of the AP.
Sum of First n Terms of an AP
The sum of the first n terms of an AP is given by Sn = n/2 [2a + (n-1)d] or equivalently Sn = n/2 (a + l), where l is the last term (nth term). The first formula is used when d is known, and the second when the last term is known. The nth term can also be found from the sum formula: an = Sn - S(n-1).
Key Points
- •Sn = n/2 [2a + (n-1)d] — use when d is known
- •Sn = n/2 (a + l) — use when the last term l is known
- •nth term from sum: an = Sn - S(n-1) for n >= 2
- •Sum of first n natural numbers = n(n+1)/2
- •If Sn is given as a formula in n, then an = Sn - S(n-1)
Worked Example
Find the sum of first 15 terms of the AP: 5, 10, 15, 20, ... a = 5, d = 5, n = 15 S15 = 15/2 [2(5) + (15-1)(5)] = 15/2 [10 + 70] = 15/2 x 80 = 600
Watch Out
If the question gives you a formula for Sn (like Sn = 3n^2 + 5n), find an by computing Sn - S(n-1). Always check a1 separately.
Quick Summary
- ✓AP: a, a+d, a+2d, ... with constant common difference d
- ✓nth term: an = a + (n-1)d
- ✓Sum of n terms: Sn = n/2 [2a + (n-1)d] = n/2 (a + l)
- ✓Three numbers are in AP if middle = average of other two
- ✓an = Sn - S(n-1) for n >= 2
- ✓Common difference d = (any term - previous term)
Key Formulas
nth term: an = a + (n-1)d
Sum of n terms: Sn = n/2 [2a + (n-1)d]
Sum of n terms (alternate): Sn = n/2 (first term + last term)
Common difference: d = (an - a)/(n-1)
nth term from sum: an = Sn - S(n-1)
Sum of first n natural numbers: n(n+1)/2
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