# Calculating the number of terms in sequences

1. Nov 13, 2013

### Cheung

How does one calculate the number of terms in the sequence

\sum\limits_{a=2}^k \sum\limits_{b=a}^k of 1/(a*b).

2. Nov 13, 2013

### Staff: Mentor

$$\sum_{a = 2}^k \sum_{b = a}^k \frac 1 {ab}$$
The "ath" term has k- a+ 1= k+1- a terms so there are $$\sum_{a= 2}^k (k+ 1- a)$$ We can write that as $$\sum_{a= 2}^k (k+1)- \sum_{a= 2}^k a$$. Of course, $$\sum{a= 2}^k (k+1)= (k-1)(k+1)$$. What is $$\sum_{a=2}^k a$$?