What will be the sum of the first 20 terms?

Step-by-step solution:

1. Identify the general term. The terms are 1/(5·10), 1/(10·15), 1/(15·20), … so the nth term is 1/(5n(5n+5)).

2. Simplify the general term: 1/(5n(5n+5)) = 1/(25 n(n+1)) = (1/25) · 1/(n(n+1)).

3. Use partial fractions: 1/(n(n+1)) = 1/n - 1/(n+1). Therefore the nth term equals (1/25)(1/n - 1/(n+1)).

4. Sum the first 20 terms. The series telescopes:

5. S20 = (1/25) · Σ_{n=1}^{20} (1/n - 1/(n+1)) = (1/25) · (1 - 1/21) = (1/25) · (20/21).

6. Compute the fraction: (1/25)·(20/21) = 20/525 = 4/105.

Final answer: 4/105