Question

If $a_1=4$ and $a_{n+1}=a_n+4n$ for $n\ge 1$, then the value of $a_{100}$ is

• A. $19804$
• B. $18904$
• C. $18894$
• D. $19904$
• E. $19894$

Answer 1

The correct option is A $19804$

Given, $a_{n+1}=a_n+4n$ and $a_1=4$
$\Rightarrow a_{n+1}-a_n=4n$
Put $n=1,a_2-a_1=4\cdot 1$
Put $n=2,a_3-a_2=4\cdot 2$
Put $n=3,a_4-a_3=4\cdot 3$
_____ ______ _________
_____ ______ _________
Put $n=99,a_{100}-a_{99}=4\cdot 99$
On adding, we get
$a_{100}-a_1=4(1+2+.....+99)$
$\Rightarrow a_{100}-4=4\frac{99(99+1)}{2}[\because \sum n=\frac{n(n+1)}{2}]$
Therefore, $a_{100}=19804$