Question

Consider a sequence whose sum to $n$ terms is given by the quadratic function ${S_n} = 3{n^2} + 5n$. Then, the sum of the squares of the first three terms of the given series is
(a) 1100
(b) 660
(c) 799
(d) 1000

Answer 1

Here, we need to find the sum of the squares of the first three terms of the given series. We will substitute the values of $n$ in the given formula for sum to find the first three terms. Then, we will add the squares of the terms to find the required sum.

Complete step-by-step answer:
Now, we will find the sum of the first term, the sum of the first two terms, and the sum of the first three terms of the series.
Substituting $n = 1$ in the given sum ${S_n} = 3{n^2} + 5n$, we get
$\Rightarrow {S_1} = 3{\left( 1 \right)^2} + 5\left( 1 \right)$
Applying the exponent on the base, we get
$\Rightarrow {S_1} = 3\left( 1 \right) + 5\left( 1 \right)$
Multiplying the terms of the expression, we get
$\Rightarrow {S_1} = 3 + 5$
Adding the terms, we get
$\Rightarrow {S_1} = 8$
We know that the sum of the first one term of the series is the first term of the series itself.
Therefore, we get
First term of the series $= 8$
Substituting $n = 2$ in the given sum ${S_n} = 3{n^2} + 5n$, we get
$\Rightarrow {S_2} = 3{\left( 2 \right)^2} + 5\left( 2 \right)$
Applying the exponent on the base, we get
$\Rightarrow {S_2} = 3\left( 4 \right) + 5\left( 2 \right)$
Multiplying the terms of the expression, we get
$\Rightarrow {S_2} = 12 + 10$
Adding the terms, we get
$\Rightarrow {S_2} = 22$
Thus, the sum of the first two terms of the series is 22.
We know that the first term of the series is 8.
Therefore, we get the equation
Second term of the series $=$ Sum of first two terms of the series $-$ First term of the series
Substituting the values, we get
Second term of the series $= 22 - 8$
Subtracting the terms, we get
Second term of the series $= 14$
Substituting $n = 3$ in the given sum ${S_n} = 3{n^2} + 5n$, we get
$\Rightarrow {S_3} = 3{\left( 3 \right)^2} + 5\left( 3 \right)$
Applying the exponent on the base, we get
$\Rightarrow {S_3} = 3\left( 9 \right) + 5\left( 3 \right)$
Multiplying the terms of the expression, we get
$\Rightarrow {S_3} = 27 + 15$
Adding the terms, we get
$\Rightarrow {S_3} = 42$
Thus, the sum of the first three terms of the series is 42.
We know that the first term of the series is 8 and the second term of the series is 14.
Therefore, we get the equation
Third term of the series $=$ Sum of first three terms of the series $-$ First term $-$ Second term
Substituting the values, we get
Third term of the series $= 42 - 8 - 14$
Subtracting the terms, we get
Third term of the series $= 20$
Therefore, we get the first three terms of the series as 8, 14, 20.
Now, we will find the sum of the squares of the first three terms of the series.
The square of 8 is 64.
The square of 14 is 196.
The square of 20 is 400.
Thus, we get
Sum of squares of first three terms of the series $= 64 + 196 + 400$
Adding the terms, we get
Sum of squares of first three terms of the series $= 660$
Therefore, we get the sum of the squares of the first three terms of the series as 660.
Thus, the correct option is option (b).

Note: We found the first three terms of the series by substituting $n$ as 1, 2, and 3. This is because the number of terms in a series cannot be 0 or less, it has to be a positive integer. Therefore, the number of terms $n$ is always a natural number, and the first three natural numbers are 1, 2, and 3.
Natural numbers include only positive numbers whereas whole numbers includes 0 along with the positive numbers. However, integers are the numbers which include both negative and positive numbers along with 0.