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Cheat sheet · JEE Preparation Mathematics

Sequences and Series cheat sheet

The whole chapter on one page: key formulas, facts and the points to remember, for quick revision before a test.

JEE · Mathematics

Cheat sheet

Class 11 · Chapter 8Sequences and Series

AM ≥ GM for positive numbers.

1. Arithmetic Progression

an=a+(n−1)da_n = a + (n - 1)d
Sn=n2[2a+(n−1)d]S_n = \dfrac n2[2a + (n - 1)d]

AM of a and b: a+b2\dfrac{a + b}{2}.

2. Geometric Progression

an=arn−1a_n = ar^{n - 1}
Sn=a(rn−1)r−1S_n = \dfrac{a(r^n - 1)}{r - 1}S∞=a1−r, ∣r∣<1S_\infty = \dfrac{a}{1 - r},\ |r| < 1

GM of a and b: ab\sqrt{ab}.

3. AM–GM Relation

A≥GA \ge G
a,b=A±(A+G)(A−G)a, b = A\pm\sqrt{(A + G)(A - G)}

Equality when a = b.

4. Key Points

  • Three numbers in AP: a − d, a, a + d.
  • Three numbers in GP: a/r, a, ar.
  • Each term of a GP × k is a GP.
  • Infinite GP needs |r| < 1.

Remember: AP & GP nth term and sum | S∞ | AM ≥ GM

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