Question

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

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

Answer 1

Answer: (C)

Given sequence is $1,2,2,3,3,3,4,4,4,4,\dots$
First term $=1$
Second term $=2$
Fourth term $=3$
Seventh term $=4$
Eleventh term $=5\dots$, so on
$\therefore$ Let $S=1+2+4+7+11\dots n$ terms
$\frac{-S=1+2+4+7+11\dots +n\text{ terms }}{0=(1+1+2+3+4\dots n\text{ terms })-a_n}$
$\Rightarrow a_n=1+\{1+2+3+4\dots (n-1)$ terms
$\Rightarrow a_n=1+\frac{n(n-1)}{2}=\frac{n^2-n+2}{2}\dots (i)$
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