Question

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

$18,15\frac{1}{2},13,...,-47$

Answer 1

Step 1. Find the common difference

The given arithmetic series is,

$18,15\frac{1}{2},13,...,-47$

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

Arithmetic progression or sequence is a mathematical sequence in which the difference between two consecutive terms is always a constant. It is represented as AP.

The difference between any two consecutive numbers is called a common difference and it is denoted by $\mathrm{d}$.

From the given,

$\mathrm{a}=18$,

So,

$\mathrm{d}=15\frac{1}{2}-18$

$=\frac{31-36}{2}$

$\therefore \mathrm{d}=\frac{-5}{2}$

Step 2. Find the number of terms in the given Arithmetic Progression.

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 the first term, $\mathrm{n}$ is the number of terms in the sequence and $\mathrm{d}$ is common difference.

${\mathrm{T}}_{\mathrm{n}}=-47$

$\mathrm{d}=\frac{-5}{2}$

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

$\Rightarrow$ $-47=18+\left(\mathrm{n}-1\right)\left(\frac{-5}{2}\right)$

$\Rightarrow$ $-47=18-\frac{5\mathrm{n}}{2}+\frac{5}{2}$

$\Rightarrow$ $\frac{5\mathrm{n}}{2}=47+18+\frac{5}{2}$

$\Rightarrow$ $\frac{5\mathrm{n}}{2}=\frac{94+36+5}{2}$

$\Rightarrow$ $5\mathrm{n}=135$

$\Rightarrow$ $\mathrm{n}=27$

Hence, the number of terms in the given arithmetic progression is $27$.