Find which of the terms in Arithmetic progression (A.P) 20, 30, 40, …. is 100 using proper formula.
(a). 11
(b). 12
(c). 13
(d). 9

- Hint: Here, we need to know the formula of finding ${{n}^{th}}$ term in the series of Arithmetic progression (A.P) which is given by ${{T}_{n}}=a+\left( n-1 \right)d$ where d is the common difference between two consecutive number, a is the first term in the series, which can be understood from the question as 20, ${{T}_{n}}$ is the term for which we need to find the value of n, which is 100.

Complete step-by-step solution -

Now, we are given a list of A.P as 20, 30, 40, ….. and we are supposed to find which number of terms is 100 in this sequence.
So, we can use the formula of ${{n}^{th}}$ term i.e. ${{T}_{n}}=a+\left( n-1 \right)d$
We have ${{T}_{n}}=100$ , $a=20$ , $d={{2}^{nd}}term-{{1}^{st}}term=30-20=10$ so, we get the common difference as 10.
Substituting all the values in the formula, we get
${{T}_{n}}=100=20+\left( n-1 \right)10$
$100=20+10n-10$
$100=10+10n$
$100-10=10n$
$90=10n$
$n=\dfrac{90}{10}=9$
Therefore, the 9th term in the sequence will be 100.
Hence, option (d) is correct.

Note: There are chances of making mistakes in interchanging the formula of finding term and summation of term i.e. ${{S}_{n}}=\dfrac{n}{2}\left( a+l \right)$ where l is the last term in the series. By solving this sum with this formula the answer will be totally different and can get confused. So, be careful when to use this formula and when to use nth term formula i.e. ${{T}_{n}}=a+\left( n-1 \right)d$ .