Sequence and Series | Mathematics for NDA PDF Download

Arithmetic Sequence and Series Formulas

Consider the arithmetic sequence a, a+d, a+2d, a+3d, a+4d, ...., where 'a' is its first term and 'd' is its common difference. Then:

• n^th term of arithmetic sequence, a_n = a + (n - 1)d
• Sum of the arithmetic series, S_n = n/2 (2a + (n - 1) d) (or) S_n = n/2 (a + an)

Geometric Sequence and Series Formulas

Consider the geometric sequence a, ar, ar², ar³, ..., where 'a' is the first term and 'r' is the common ratio. Then:

• nth term of geometric sequence, a_n = a r^{n - 1}
• Sum of the finite geometric series (sum of first 'n' terms), S_n = a (1 - rⁿ) / (1- r)
• Sum of infinite geometric series, S_n = a / (1 - r) when |r| < 1. The sum is not defined when |r| ≥ 1.

Harmonic Sequence and Series Formulas

Consider the harmonic sequence 1/a, 1/(a+d), 1/(a+2d), 1/(a+3d), 1/(a+4d), ...., where '1/a' is its first term and 'd' is the common difference of the arithmetic sequence a, a + d, a + 2d, .... Then:

• n^th term of harmonic sequence, an = 1 / (a + (n - 1) d)
• Sum of the harmonic series, S_n = 1/d in [(2a + (2n - 1) d) / (2a - d)]