CBSE Class 10 Mathematics Arithmetic Progressions Notes Set 10

Arithmetic Progressions: Key Ideas

An arithmetic progression (AP) is a sequence in which every term is obtained from the previous one by adding the same number d, the common difference. If the first term is a, the terms are a, a + d, a + 2d, ...
General term: \( a_n = a + (n-1)d \). The term from the end of a finite AP with last term l is l − (n − 1)d.
Three numbers in AP are best taken as a − d, a, a + d; four numbers as a − 3d, a − d, a + d, a + 3d.
A sequence is an AP exactly when a_{n+1} − a_n is the same for every n. The middle term is the average of its two neighbours: 2b = a + c.

Sum of an AP

Sum of the first n terms: \( S_n = \\frac{n}{2}\\,[2a + (n-1)d] = \\frac{n}{2}(a + l) \).
The nth term can be recovered from the sum: \( a_n = S_n - S_{n-1} \) for n ≥ 2, and a_1 = S_1.
If S_n = An² + Bn then the sequence is an AP with common difference 2A and a_n = 2An + (B − A).
Sum of the first n natural numbers is n(n + 1)/2, of the first n odd numbers is n², and of the first n even numbers is n(n + 1).

Arithmetic Means

Inserting k arithmetic means between a and b gives an AP with k + 2 terms, so the common difference is d = (b − a)/(k + 1).
The sum of the k inserted means is k(a + b)/2, that is k times the single mean of a and b.
The arithmetic mean of a and b is (a + b)/2.

Useful Results

If the pth term is q and the qth term is p, then a_{p+q} = 0.
If the sum of p terms equals the sum of q terms (p ≠ q), then the sum of p + q terms is zero.
If the mth term is 1/n and the nth term is 1/m, then a = d = 1/(mn) and a_{mn} = 1.
If the ratio of the sums of n terms of two APs is f(n), the ratio of their mth terms is f(2m − 1).
If a, b, c are in AP then 1/(b + c), 1/(c + a), 1/(a + b) are in AP exactly when a², b², c² are in AP.
Counting multiples: the number of multiples of k between two limits is found by locating the first and last multiple and using n = (last − first)/k + 1.