Question

Evaluate

$0.5+0.51+0.52+....$to $100$ terms.

Answer 1

Step 1. Note the given data:

The given series $0.5+0.51+0.52+....$ to $100$ is an A.P. with

First term $a=0.5$

Number of terms $n=100$

Common difference is $0.51-0.5=0.01$

Step 2. Find the sum of the first $100$ terms of a given A.P.

The formula of the sum of the $n$ term of an A.P. is $S_n=\frac{n}{2}\left(2a+\left(n-1\right)d\right)$

Where $a$ is the first term and $d$ is common difference.

Here $n=100,a=0.5,d=0.01$

Substituting the value

$$\begin{gathered} S_{100}=\frac{100}{2}\left(2\left(0.5\right)+\left(100-1\right)0.01\right) \\ {S}_{100}=50\left(1+\left(99\right)0.01\right) \\ {S}_{100}=50\left(1+0.99\right) \\ {S}_{100}=50\left(1.99\right) \\ {S}_{100}=99.5 \end{gathered}$$

Hence, the sum of first $100$ terms of the given A.P. is $99.5$.