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08Class 11 Maths

Sequences and Series formulas

AP, GP and series sum formulas. Questions often give you a mix of terms and sums, so learn to write everything using a and d, or a and r.

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All Maths formulas

Arithmetic progression

nth term
an=a+(n−1)d\displaystyle a_n = a + (n - 1)d
Sum of n terms
Sn=n2[2a+(n−1)d]=n2(a+l)\displaystyle S_n = \frac{n}{2}[2a + (n - 1)d] = \frac{n}{2}(a + l)

l is the last term.

Arithmetic mean
A=a+b2\displaystyle A = \frac{a + b}{2}

Geometric progression

nth term
an=arn−1\displaystyle a_n = ar^{n - 1}
Sum of n terms
Sn=a(rn−1)r−1\displaystyle S_n = \frac{a(r^n - 1)}{r - 1}

For r ≠ 1.

Sum to infinity
S∞=a1−r\displaystyle S_\infty = \frac{a}{1 - r}

Only when |r| < 1.

Geometric mean
G=ab\displaystyle G = \sqrt{ab}
AM and GM
A≥G\displaystyle A \ge G

For positive numbers.

Standard sums

First n natural numbers
∑k=n(n+1)2\displaystyle \sum k = \frac{n(n + 1)}{2}
Squares
∑k2=n(n+1)(2n+1)6\displaystyle \sum k^2 = \frac{n(n + 1)(2n + 1)}{6}
Cubes
∑k3=[n(n+1)2]2\displaystyle \sum k^3 = \left[\frac{n(n + 1)}{2}\right]^2

Where students slip up

  • Using the infinite GP formula when |r| ≥ 1. The series doesn't converge.
  • Confusing n terms with the nth term, since Sₙ and aₙ are different things.

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