CBSE · Class 10 · Mathematics · Chapter 5

Arithmetic Progressions — Formula Sheet

Board Formulas

8 formulas

1.General Term (nth term)★
$a_{n} = a + (n - 1) d$
$a$ : First term of the AP · $d$ : Common difference (a₂ − a₁ = a₃ − a₂) · $n$ : Position (natural number)
a = first term, d = common difference. Find nth term without listing all previous terms.
2.Common Difference
$d = a_{2} - a_{1} = a_{3} - a_{2} = \dots$
Constant difference between consecutive terms defines an AP.
3.Sum of First n Terms★
$S_{n} = \frac{n}{2} [2 a + (n - 1) d]$
Use when 'd' is known but the last term is not.
4.Sum when Last Term Is Known★
$S_{n} = \frac{n}{2} (a + l)$
l = last term = aₙ. Use when a and l are given.
5.Middle-term (Three in AP)
$2 b = a + c$
a, b, c are in AP iff b is the arithmetic mean of a and c.
6.n-th Term from End
$a_{n}^{'} = l - (n - 1) d$
l = last term. Useful to find, say, '5th term from the end'.
7.Number of Terms n from Given aₙ
$n = \frac{a _{n} - a}{d} + 1$
Rearranged nth-term formula. n MUST be a positive integer for a valid AP term.
8.Standard Sums
$\sum_{k = 1}^{n} k = \frac{n ( n + 1 )}{2}, \sum_{k = 1}^{n} (2 k - 1) = n^{2}$
Sum of first n natural numbers and first n odd numbers. Useful shortcuts.