# GATE Papers >> Civil >> 2018 >> Question No 9

Question No. 9

Consider a sequence of numbers $\style{font-family:'Times New Roman'}{a_1,a_2,\;a_3,......,\;a_n}$ where $a_n=\frac1n-\frac1{n+2}$ for each integer $\style{font-family:'Times New Roman'}{n>0}$. What is the sum of the first 50 terms?

##### Answer : (C) $\style{font-family:'Times New Roman'}{\left(1+\frac12\right)-\left(\frac1{51}+\frac1{52}\right)}$

Solution of Question No 9 of GATE 2018 Civil Paper

a1 + a2 + a3 +...+ a50

$\begin{array}{l}=\lbrack\left(1-\frac13\right)+\left(\frac12-\frac14\right)+\left(\frac13-\frac15\right)+\left(\frac14-\frac16\right)+\left(\frac15-\frac17\right)+...\\\;\;+\left(\frac1{47}-\frac1{49}\right)+\left(\frac1{48}-\frac1{50}\right)+\left(\frac1{49}-\frac1{51}\right)+\left(\frac1{50}-\frac1{52}\right)\rbrack\\=1+\frac12-\frac1{51}-\frac1{52}\\=\left(1+\frac12\right)-\left(\frac1{51}+\frac1{52}\right)\\\end{array}$