Question

Find the sum of following series: \[7 + 10.5 + 14 + ..... + 84\]
A) \[1046.5\]
B) \[3455.3\]
C) \[1466\]
D) \[1346\]

Answer 1

Hint:
Here, we will use the concept of A.P. as the series given is in A.P. form. Firstly we will find the common difference of the series. Then we will find the number of terms in the series. We will use the formula of the sum of \[n\] number of terms in an A.P. series to get the required answers.

Formula used:
We will use the formula, sum of \[n\] number of terms \[ = \dfrac{n}{2}\left( {a + {n^{th}}term} \right)\]

Complete step by step solution:
The series given is \[7 + 10.5 + 14 + ..... + 84\].
We can see that the series given is in arithmetic progression form. The first term \[a = 7\] and we have to find the common difference.
So,
Common difference, \[d = 10.5 - 7 = 3.5\].
Now we have to find the number of terms in the given series. We know that the \[{n^{th}}\] is \[84\]. Also the formula of \[{n^{th}}\] term is \[{n^{th}}{\rm{term}} = a + \left( {n - 1} \right)d\].
Therefore, we get
\[ \Rightarrow {n^{th}}{\rm{term}} = a + \left( {n - 1} \right)d = 84\]
\[ \Rightarrow 7 + \left( {n - 1} \right)3.5 = 84\]
Subtracting 7 from both the sides, we get
\[\begin{array}{l} \Rightarrow \left( {n - 1} \right)3.5 = 84 - 7\\ \Rightarrow \left( {n - 1} \right)3.5 = 77\end{array}\]
Dividing both side by \[3.5\], we get
\[ \Rightarrow \left( {n - 1} \right) = \dfrac{{77}}{{3.5}} = 22\]
Adding 1 both the sides, we get
\[ \Rightarrow n = 22 + 1 = 23\]
Hence, there are 23 terms in the given series of number.
Now, we will find the sum of the series by using the formula of the sum of \[n\] number of terms in an A.P. series.
We know that the sum of the \[n\] number of terms of an A.P. \[ = \dfrac{n}{2}\left( {a + {n^{th}}term} \right)\]. Therefore, we get
\[ \Rightarrow 7 + 10.5 + 14 + ..... + 84 = \dfrac{{23}}{2}\left( {7 + 84} \right)\]
Adding the terms in the bracket, we get
\[ \Rightarrow 7 + 10.5 + 14 + ..... + 84 = \dfrac{{23}}{2}\left( {91} \right)\]
Multiplying the terms, we get
\[ \Rightarrow 7 + 10.5 + 14 + ..... + 84 = \dfrac{{2093}}{2}\]
Dividing the term by 2, we get
\[ \Rightarrow 7 + 10.5 + 14 + ..... + 84 = 1046.5\]
Hence, \[1046.5\] is the sum of the given number series.

So, option A is the correct option.

Note:
We used the terms arithmetic progression and common difference in the solution.
An arithmetic progression is a series of numbers in which each successive number is the sum of the previous number and a fixed difference. The fixed difference is called the common difference.
We calculated the common difference by subtracting the first term from the second term. The common difference is the fixed difference between each successive term of an A.P.
Therefore, we get
Common difference \[ = \] Second term \[ - \] First term \[ = \] Third term \[ - \] Second term \[ = \] Fourth term \[ - \] Third term