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Formula Sheet

Sequence & Series Formulas

All key formulas grouped by subtopic. Each one has a quick reminder and common mistakes to watch for.

10 formulas · 7 subtopics

nth Term of AP

#1
an=a+(n−1)da_n = a + (n-1)d

💡 a = first term, d = common difference. Works for any integer n.

⚠ Using nd instead of (n-1)d.

Sum of n Terms of AP

#2
Sn=n2[2a+(n−1)d]=n2(a+l)S_n = \frac{n}{2}[2a + (n-1)d] = \frac{n}{2}(a + l)

💡 l = last term. Second form is faster when you know the last term.

⚠ Confusing S_n with a_n. S_n is cumulative, a_n is a single term.

nth Term of GP

#3
an=arn−1a_n = ar^{n-1}

💡 a = first term, r = common ratio. Valid for all positive integers n.

Sum of n Terms of GP

#4
Sn=a⋅rn−1r−1,r≠1S_n = a\cdot\frac{r^n - 1}{r - 1}, \quad r \neq 1

💡 Use (1-r^n)/(1-r) when |r|<1 to avoid sign confusion.

⚠ Using this formula when r=1. If r=1, S_n = na.

Sum of Infinite GP

#5
S∞=a1−r,∣r∣<1S_\infty = \frac{a}{1-r}, \quad |r| < 1

💡 Only converges when |r|<1. This is the most tested GP formula in JEE.

Sum of First n Natural Numbers

#6
∑k=1nk=n(n+1)2\sum_{k=1}^{n} k = \frac{n(n+1)}{2}

💡 Building block for all summation formulas.

Sum of Squares

#7
∑k=1nk2=n(n+1)(2n+1)6\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}

💡 Frequently appears in method-of-differences and series summation.

Sum of Cubes

#8
∑k=1nk3=[n(n+1)2]2\sum_{k=1}^{n} k^3 = \left[\frac{n(n+1)}{2}\right]^2

💡 Sum of cubes = (sum of first n numbers) squared. Elegant identity.

Sum of AGP

#9
Sn=a1−r+dr(1−rn−1)(1−r)2−(a+(n−1)d)rn1−rS_n = \frac{a}{1-r} + \frac{dr(1-r^{n-1})}{(1-r)^2} - \frac{(a+(n-1)d)r^n}{1-r}

💡 Multiply S by r, subtract from S to reduce the AP part. Works every time.

⚠ Not recognizing an AGP. Look for products of linear and geometric terms.

Method of Differences

#10
Tn=Sn−Sn−1,then find Tn pattern and sumT_n = S_n - S_{n-1}, \quad \text{then find } T_n \text{ pattern and sum}

💡 If first differences form an AP or GP, use this to find the general term.