Question

How do you find the 20th term and the sum of 20 first terms of the sequence 2, 5, 8, 11, 14, 17……?

Answer 1

Hint: In these types of questions which the given series is in arithmetic series we will make use of sum of $n$ terms in the arithmetic series formula that can be can be written as, ${S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]$, and also we will make use of the formula for number of terms in an arithmetic series i.e., .., by substituting the values in the formulas we will get the required answers.

Complete step by step answer:
Given sequence is 2, 5, 8, 11, 14, 17……,
The given sequence is in arithmetic progression as it has same common difference which is 3,
So, now the sum of $n$ terms in an arithmetic series can be written as,
${S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]$,
Here we have to fine the sum of 20 terms, so here $n = 20$,$a = 2$, $d = 3$, by substituting the value in the formulas we get,
 $ \Rightarrow {S_{20}} = \dfrac{{20}}{2}\left[ {2\left( 2 \right) + \left( {20 - 1} \right)3} \right]$,
Now simplifying we get,
$ \Rightarrow {S_{20}} = 10\left[ {4 + \left( {19} \right)3} \right]$,
Now multiplying the terms inside the brackets we get,
$ \Rightarrow {S_{20}} = 10\left[ {4 + 57} \right]$,
Now again simplifying we get,
$ \Rightarrow {S_{20}} = 10\left[ {61} \right]$,
Now multiplying to simplify we get,
$ \Rightarrow {S_{20}} = 610$,
So, the sum of 20 terms is 610,
Now we have to find the 20th term of the given sequence, we know The ${n^{th}}$ term In A.P is given by ${T_n} = a + \left( {n - 1} \right)d$,
By substituting the values in the formula we get, here $n = 20$,$a = 2$, $d = 3$, we get,
$ \Rightarrow {T_{20}} = 2 + \left( {20 - 1} \right)\left( 3 \right)$,
Now simplifying we get,
$ \Rightarrow {T_{20}} = 2 + \left( {19} \right)\left( 3 \right)$,
Now multiplying we get,
$ \Rightarrow {T_{20}} = 2 + 57$,
Now further simplification we get,
$ \Rightarrow {T_{20}} = 59$,
So, the 20th term is 59.

$\therefore $ The 20th term of the given sequence 2, 5, 8, 11, 14, 17…… is 59, and the sum of 20 terms of the sequence 2, 5, 8, 11, 14, 17…… is 610.

Note: As there are 3 types of series i.e., Arithmetic series, Geometric series and Harmonic series, students should not get confused which series to be used in what type of questions, as there are many formulas related to each of the series, here are some useful formulas related to the above series,
Sum of the $n$ terms in A. P is given by, ${S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]$, where $n$ is common difference, $a$ is the first term.
The ${n^{th}}$ term In A.P is given by ${T_n} = a + \left( {n - 1} \right)d$,
Sum of the $n$ terms in GP is given by,${S_n} = \dfrac{{a\left( {1 - {r^n}} \right)}}{{1 - r}}$, where $r$ is common ratio, $a$ is the first term.
The ${n^{th}}$ term In G.P is given by ${T_n} = a{r^{n - 1}}$,
If a, b, c are in HP, then b is the harmonic mean between a and c.
In this case, $b = \dfrac{{2ac}}{{a + c}}$.