Question

Find the sum of the series \(\rm \frac{1}{4}+\frac{1}{2}+\frac{3}{4}+1+\frac{10}{8}+....\)up to n terms.
1. n(n + 1)/4
2. n(n + 1)/2
3. n(n + 1)
4. n(n + 1)/8
5. None of these

Answer 1

Correct Answer - Option 4 : n(n + 1)/8

Concept:

• Sum of the first n even natural numbers i.e 2 + 4 + 6 +8 +10 + ...up to n terms = n(n + 1)
• Sum of the first n odd natural numbers i.e 1 + 3 + 5 +7 +9 + ...up to n terms = n2 

Calculation:

Let the nth term and sum of the series upto n terms of the series be T_n and S_n, respectively. Then,

\(\rm S_n = \frac{1}{4}+\frac{1}{2}+\frac{3}{4}+1+\frac{10}{8}+....\)

Multiplying the above equation by 8, we get

⇒ 8S_n = 2 + 4 + 6 + 8 + 10 +.......up to n terms

This is the sum of first even natural numbers.

As we know that, sum of the first n even natural numbers i.e 2 + 4 + 6 +8 +10 + ...up to n terms = n(n + 1)

⇒ 8S_n = n(n + 1) ⇒ Sn = n(n + 1)/8

Hence, option 4 is the correct answer.