Question

The formula to find $n$ th term of an AP is
A). $a + \left( {n - 1} \right)d$
B). $a{r^{n - 1}}$
C). $\dfrac{1}{{a + \left( {n - 1} \right)d}}$
D). $a - \left( {n - 1} \right)d$

Answer 1

Hint: In an arithmetic progression the first term is given as $a$ and the common difference of the AP is given as $d$ . The consecutive terms of AP differ by a common difference $d$. The next term of a given AP is given by adding the common difference to the previous term.

Complete step-by-step solution:
In an arithmetic progression , the consecutive terms differ from each other by a common difference
The first term of an AP is usually represented by $a$ .
The common difference of the AP is represented by $d$ .
The $n$ th term of an AP is represented as ${t_n}$
The $n$ th term of an AP is given by adding to the first term $n - 1$ times the common difference of the AP
From this, we get the formula of the $n$ th term of an AP to be $a + \left( {n - 1} \right)d$
Therefore, the correct option is option (A).

Additional Information:
The sum upto $n$ terms of an arithmetic progression is given by the formula
$\bullet$ ${S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]$ , where ${S_n}$ is the sum upto $n$ terms , $n$ is the number of terms , $a$ is the first term and $d$ is the common difference.
$\bullet$ In an AP , the $n$ th term ${t_n}$ and the last term $l$ are equal when $n$ is the total number of terms in the AP

Note: Considering the other options , the second option is the formula of the $n$ th term of the geometric progression . The third option is not correct as it the reciprocal of the formula $n$ th term of an AP .The fourth option is not correct as the $n$ th term of an AP is given by adding to the first term $n - 1$ times the common difference of the AP but in this option it is given as subtracting $n - 1$ times the common difference of the AP from the first term .