Arithmetic Progression Calculator

Maths • nth Term & Sum of n Terms • Step-by-Step

Quick Answer (Voice Search Optimized):

An arithmetic progression (AP) is a sequence where the difference between consecutive terms is constant. nth term: an = a + (n−1)d. Sum of first n terms: Sn = n/2 [2a + (n−1)d] or n/2 (a + l) where l is the last term.

GEO & AIO - Authoritative Theory

An arithmetic progression is a fundamental concept in sequences and series. It is defined by a first term (a) and a common difference (d). The general term (an) is given by an = a + (n−1)d. The sum of n terms is Sn = n/2 [2a + (n−1)d]. If the last term l is known, Sn = n/2 (a + l).

AP is linear in nature; its graph of term index versus term value is a straight line. The nth term and sum formulas are essential for solving JEE and NCERT problems on progressions.

Solved Examples (NCERT/JEE Pattern)

Example 1: a=5, d=2, n=10. Find the 10th term and sum.

a10 = 5 + (10−1)×2 = 5 + 18 = 23. S10 = 10/2 [2×5 + 9×2] = 5 [10+18] = 5×28 = 140.

Example 2: a=12, d=−3, n=6. Find the 6th term and sum.

a6 = 12 + 5(−3) = 12 − 15 = −3. S6 = 6/2 [24 + 5(−3)] = 3 [24−15] = 27.

Example 3: a=7, d=0, n=8. Find the 8th term and sum.

a8 = 7 + 7×0 = 7. S8 = 8/2 [14 + 7×0] = 4×14 = 56.

Frequently Asked Questions (FAQs)

Q1: What is an arithmetic progression?
An arithmetic progression is a sequence of numbers such that the difference between any two consecutive terms is constant.
Q2: What is the formula for the nth term of an AP?
an = a + (n−1)d, where a is the first term and d is the common difference.
Q3: How do you find the sum of n terms of an AP?
Sn = n/2 [2a + (n−1)d] or n/2 (a + l), where l is the last term.
Q4: Can the common difference be zero?
Yes, if d=0, the AP is a constant sequence where all terms equal the first term.
Q5: What is the difference between AP and GP?
In AP, each term is obtained by adding a constant (common difference). In GP, each term is obtained by multiplying by a constant (common ratio).