Question

If the sum of the first $14$ terms of $AP$ is $1050$ and its first term is $10$, find the $20{\text{th}}$ term.

Answer 1

Hint: Here we will simply apply the formula of the sum of the $n$ terms of $AP$ which is the Arithmetic progression and get the value of the common difference by this formula. Once we get the common difference $d$ then we can simply solve for $20{\text{th}}$ term by applying the formula of the $n{\text{th}}$ term of $AP$

Formula Used:
${\text{sum of n terms}}({S_n}) = \dfrac{n}{2}(2a + (n - 1)d)$
$n{\text{th term}} = {a_n} = a + (n - 1)d$

Complete step-by-step answer:
AP or arithmetic progression is the sequence in which the different terms have the same common difference or we can say that the consecutive numbers differ by the same number. For example: in the sequence like $2,4,6,8,.......100$ we can see that the difference between each consecutive term is $2$ as $4 - 2 = 6 - 4 = 8 - 6 = 2$
Hence the given sequence is called the Arithmetic progression or AP
Here we are given that the sum of first $14$ terms of$AP$ is $1050$
So we can apply the formula of the sum of the $n$ terms which is
${\text{sum of n terms}}({S_n}) = \dfrac{n}{2}(2a + (n - 1)d)$
We know that
${\text{Sum}} = 1050$
$a = {\text{first term}} = 10$
$n = {\text{number of terms}} = 14$
Here $d = $common difference
So substituting the values in the formula we get:
${\text{sum of n terms}}({S_n}) = \dfrac{n}{2}(2a + (n - 1)d)$
$\Rightarrow$ $1050 = \dfrac{{14}}{2}(2(10) + (14 - 1)d)$
$\Rightarrow$ $1050 = 7(20 + 13d)$
$\Rightarrow$ $\dfrac{{1050}}{7} = 20 + 13d$
$\Rightarrow$ $150 = 20 + 13d$
$\Rightarrow$ $13d = 130$
$\Rightarrow$ $d = 10$
Hence we get that the common difference of the given arithmetic progression is $10$ which means that each term of the given sequence is $10$ more than the previous one.
Now we have got the common difference and now we need to know the $20{\text{th}}$ term of the sequence.
So we apply the formula of the $n{\text{th}}$ term of the AP we get
$n{\text{th term}} = {a_n} = a + (n - 1)d$
Here we should know what the value of each variable in the question is:
${a_n} = n{\text{th term}}$
And $a = $first term$ = 10$
And $n = 20$ as we need to find the $20{\text{th}}$ term
And $d = 10$ as we had calculated earlier
${a_n} = 10 + (20 - 1)10$
$
   = 10 + (19)(10) \\
   = 10 + 190 \\
   = 200 \\
 $
Hence we get that the $20{\text{th}}$ term of the sequence which is in AP is $200$.

Note: Here we need to understand the meaning of the Arithmetic progression and we should know what formula should be used in order to calculate the $n{\text{th}}$ term and the sum of the n terms. We should not make calculation mistakes as these types of questions are simple but need just the formula and the values of the parameters used in the formula.