Maharashtra State Board · Class 10 · Mathematics · Algebra Chapter 3

Arithmetic Progression — Formula Sheet

Board Formulas

10 formulas

1.nth Term of AP★
$a_{n} = a + (n - 1) d$
$a_{n}$ : nth term of the AP · $a$ : First term · $d$ : Common difference · $n$ : Position of the term
a = first term, d = common difference, n = position.
2.Sum of First n Terms★
$S_{n} = \frac{n}{2} [2 a + (n - 1) d]$
Also written as S_n = (n/2)(a + l), where l is the last term.
3.Sum in Terms of First and Last
$S_{n} = \frac{n}{2} (a + l)$
Use when last term l is known — the quickest route.
4.Common Difference from Two Terms
$d = a_{n} - a_{n - 1}$
d is constant for a valid AP. Check at least three consecutive pairs before concluding it is an AP.
5.General Term from Two Given Terms
$d = \frac{a _{p} - a _{q}}{p - q}$
Useful when specific term positions and values are given.
6.Arithmetic Mean
$b = \frac{a + c}{2}$
If a, b, c are in AP, then b is the arithmetic mean of a and c.
7.Sum of First n Natural Numbers
$1 + 2 + 3 + \dots + n = \frac{n ( n + 1 )}{2}$
Special AP with a = 1, d = 1.
8.Sum of First n Odd Numbers
$1 + 3 + 5 + \dots + (2 n - 1) = n^{2}$
Special AP with a = 1, d = 2. Neat perfect-square pattern.
9.Sum of First n Even Numbers
$2 + 4 + 6 + \dots + 2 n = n (n + 1)$
Twice the sum of first n naturals.
10.Term from Sums
$a_{n} = S_{n} - S_{n - 1}$
If S_n formula is given, this recovers the nth term.