Question

Find the number of terms in each of the following A.P.

$7,13,19,\dots ,205$

Answer 1

Step 1. Find the common difference

The given arithmetic series is,

$7,13,19,\dots ,205$

We have to find the ${\mathrm{n}}^{\mathrm{th}}$ term of an A P.

As we know that the arithmetic progression is a sequence of numbers such that the difference d between consecutive term is a constant.

From the given,

$\mathrm{a}=7$,

So the common difference is

$\mathrm{d}=13-7$

$\therefore \mathrm{d}=6$

Step 2. Find the number of terms

Formula to find the${\mathrm{n}}^{\mathrm{th}}$ term of an A P. is,

${\mathbf{T}}_{\mathbf{n}}\mathbf{=}\left[a+\left(n-1\right)\times d\right]$

Here, ${\mathrm{T}}_{\mathrm{n}}$ is the ${\mathrm{n}}^{\mathrm{th}}$ term, $\mathrm{a}$ is first term, $\mathrm{n}$ is the number of terms in the sequence and $\mathrm{d}$ is a common difference.

${\mathrm{t}}_{\mathrm{n}}=205$

By using the formula, the ${\mathrm{n}}^{\mathrm{th}}$ term is,

$\Rightarrow$$205=7+\left(n-1\right)\times 6$

$\Rightarrow$ $205=6n+1$

$\Rightarrow$ $204=6n$

$\Rightarrow$ $n=34$

Hence, the number of terms is $34$.