Question

The sum of first n terms of an arithmetic sequence is $5{n^2} + 2n$ .
(a) What is the sum of the first two terms of this sequence ?
(b) Write the first two terms of the sequence .

Answer 1

Hint: As in the sum of first n terms of an arithmetic sequence is $5{n^2} + 2n$ so in part (a) put $n = 2$ then we get sum first two term in part (b) we know that the sum of the one term is the term itself hence if we put $n = 1$ in the equation we get first term for second term use ${a_n} = {S_{n + 1}} - {S_n}$ .

Complete step-by-step answer:
At in the given question it is given that the sum of first n terms of an arithmetic sequence is $5{n^2} + 2n$ where $n$ is the number of the term up to which we have to find the sum .
So in the part (a) we have to find the sum of first two term for this we have to do is ,
Put $n = 2$ so that we get the sum of first two term
Hence on putting $n = 2$ in the equation $5{n^2} + 2n$
$5{\left( 2 \right)^2} + 2\left( 2 \right)$
on solving $5 \times 4 + 4$ = $24$
Hence the sum of first two term is $24$
Now in the part (b) we have to find out the first two term of the sequence
we know that the sum of the one term is the term itself hence if we put $n = 1$ in the equation $5{n^2} + 2n$ we get the first term,
${a_1} = 5(1) + 2 \times 1$
$\Rightarrow$ ${a_1} = 7$
So the first term is $7$
As for the second term use formula ${a_n} = {S_{n + 1}} - {S_n}$ but this is not valid for $n = 1$ where
${S_n}$ = $5{n^2} + 2n$
$\Rightarrow$ ${a_2} = {S_3} - {S_2}$ now put $n = 3$ and $n = 2$ in the equation $5{n^2} + 2n$
$\Rightarrow$ ${a_2} = 5{(3)^2} + 2(3) - 5{(2)^2} - 2(2)$
On solving ,
$\Rightarrow$ ${a_2} = 45 + 6 - 20 - 4$ = $27$
Hence the second term is $27$.

Note: As in the question we can also find the sequence equation in the form of term we know that the ${a_n} = {S_{n + 1}} - {S_n}$ ( $n \ne 1$ ) and ${S_n}$ = $5{n^2} + 2n$ So on putting
${a_n} = 5{\left( {n + 1} \right)^2} + 2(n + 1) - 5{n^2} - 2n$ ,
$2n$ is cancel out , ${a_n} = 5{\left( {n + 1} \right)^2} + 2 - 5{n^2}$ on solving further ${a_n} = 5\left( {{n^2} + 2n + 1} \right) + 2 - 5{n^2}$
${a_n} = 10n + 7$ but not valid for $n = 1$