Question

If the sum of first \[n\] terms of a series be \[5{n^2} + 2n\] , then its second term is:
A) \[16\]
B) \[17\]
C) \[\dfrac{{27}}{{14}}\]
D) \[\dfrac{{56}}{{15}}\]

Answer 1

Hint: To solve the problem, we have to find the first term and the sum of the first two numbers. For that, we have to substitute n by 1 and 2 we will get the first number of the series and adding the two numbers we will get the sum of the first two numbers of the series. Then by subtracting these two values we get the second term of the series.

Complete step-by-step answer:
It is given that; the sum of first \[n\] terms of a series be \[5{n^2} + 2n\].
We have to find the second term.
To solve the sum, we have to find the first term and the sum of the first two numbers.
By substituting n by 1 and 2 we will get the first number of the series and the sum of the first two numbers of the series.
Now, substituting \[n = 1\] we get,
First term is,
$\Rightarrow$\[5{(1)^2} + 2(1) = 7\]
Again, substituting \[n = 1\] we get,
Sum of first two terms is,
$\Rightarrow$\[5{(2)^2} + 2(2) = 24\]
So, the second term is \[24 - 7 = 17\]
Hence, the second term is \[17\].

Hence, the correct option is B) \[17\].

Note: An itemized collection of elements in which repetitions of any sort are allowed is known as a sequence, whereas series is the sum of all elements.
The sum of the terms of a sequence is called a series .
If a sequence is arithmetic or geometric there are formulas to find the sum of the first n terms, denoted \[{S_n}\] , without actually adding all of the terms.
We can consider, \[{S_n} = 5{n^2} + 2n\]
\[n = 1,{S_1} = 7\]
\[n = 2,{S_1} = {5.2^2} + 4 = 24\]
So, the second term is \[{S_2} - {S_1} = 24 - 7 = 17\]
Hence, the second term is \[17\].