Question

Find the sum to n terms of the following series. 

1. 1 × 4 + 3 × 7 + 5 × 10 +……….

2. 1 × 2² + 2 × 3² + 3 × 4² +………..

3. \(\frac{1}{1\times 2} + \frac{1}{2\times 3}+\frac{1}{3\times 4}\) + .....

4. 1² + 3²+ 5⁵ + .....

Answer 1

1. The given series is the product of two AP 1 , 3, 5, with 

t_n = 1 + (n – 1)2 

= 2n – 1 4, 7, 10, with 

t_n = 4 + (n 1)3 = 3n + 1

Then the n^th term of the given series is

t_n = (2n – 1)(3n + 1) = 6n² – n – 1

2. The given series has two AP 1, 2, 3,……. with 

t_n = 1 + (n – 1)1 = n 2, 3, 4,…… with 

t_n = 2 + (n – 1)1 = n + 1

Then the n^th term of the given series is

t_n = n(n + 1)² = n(n² + 2n + 1) = n³ + 2n² + n

3.

4. The given series has an AP 1, 3, 5……… with 

t_n = 1 + (n – 1)2 = 2n – 1

Then the n^th term of the given series is

t_n = (2n – 1)² = 4n² – 4n + 1