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

Since we are given the sum of n terms of the sequence to be $5{n^2} + 2n$ and we know that ${S_1}$ is the first term of the sequence and ${S_2}$ is the sum of the first two terms of the sequence and replacing n by 1 and 2 we get the values of ${S_1}$ and ${S_2}$ and using this we can get the values of ${a_1}$ and ${a_2}$

Complete step-by-step answer:
We are given that the sum of the first n terms of the sequence is $5{n^2} + 2n$
$\Rightarrow {S_n} = 5{n^2} + 2n$ ………(1)
Now replacing 1 in the place of n we get ${S_1}$
It is clear that ${S_1}$ is the first term , that is , ${a_1}$
Hence ${S_2}$ is the sum of the first two terms of the sequence
$\Rightarrow {S_2} = {a_1} + {a_2}$
Now substituting n = 1 in (1) we get
$\Rightarrow {S_1} = 5{\left( 1 \right)^2} + 2\left( 1 \right) = 5 + 2 = 7$
And since ${S_1} = {a_1}$ we get
$\Rightarrow {a_1} = 7$……..(2)
Now lets substitute n = 2 in (1)
$\Rightarrow {S_2} = 5{\left( 2 \right)^2} + 2\left( 2 \right) = 20 + 4 = 24$
And since ${S_2} = {a_1} + {a_2}$ we get
$\Rightarrow {a_1} + {a_2} = 24$…………(3)
Hence now we get the sum of the first two terms of the sequence to be 24
Now lets substitute (2) in (3)
$\Rightarrow 7 + {a_2} = 24 \\\ \Rightarrow {a_2} = 24 - 7 = 17 \\\$
And hence now we get that the first term of the sequence to be 7 and the second term to be 17.

Additional information :
The sum of the arithmetic sequence formula is used to calculate the total of all the digits present in an arithmetic progression or series.

Note: Many students tend to use the formula of sum to n terms . It will only make the sum more difficult to solve. Arithmetic sequence is a list of numbers with a definite pattern. If we take any number in the sequence then subtract it by the previous one, and the result is always the same or constant then it is an arithmetic sequence.