Question

The ${100}^{th}$ term of the sequence $1,2,2,3,3,3,4,4,4,4,...,$ is

• A. $12$
• B. $13$
• C. $14$
• D. $15$
• E. $16$

Answer 1

The correct option is C $14$

Given sequence is $1,2,2,3,3,3,4,4,4,4,...$
First term $=1$
Second term $=2$
Fourth term $=3$
Seventh term $=4$
Eleventh term $=5...,$ so on
Let $S=1+2+4+7+11...n$ terms
$\underset{–}{{}_{-}S=1+2+4+7+11...+n\text{terms}}$
$0=(1+1+2+3+4...n\text{terms})-a_n$
$\Rightarrow a_n=1+\{1+2+3+4...(n-1\left)\text{terms}\right\}$
$\Rightarrow a_n=1+\frac{n(n-1)}{2}=\frac{n^2-n+2}{2}....\left(i\right)$
if $n=14$, then $a_n=92$
i.e., ${92}^{nd}$ term is $14$.
If $n=15$, then $a_n=106$
i.e., ${106}^{th}$ term is $15$.
Hence, ${100}^{th}$ term is $14$.