Question

How to do you find the sum of the arithmetic sequence $4 + 7 + 10 + ..... + 22?$

Answer 1

This problem deals with both arithmetic and geometric progression. In an arithmetic progression, the consecutive terms differ by a common difference $d$, with an initial term $a$ at the beginning. The following formulas are used in the problem.
The sum of the $n$ terms in an A.P. is given by:
$\Rightarrow {S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]$

Complete step-by-step solution:
Given an arithmetic progression as $4,7,10,.....,22$
The initial term $a$, of the above given arithmetic progression is given by:
$\Rightarrow a = 4$
The common difference $d$, of the given arithmetic progression is given by:
$\Rightarrow d = 3$
Here to find out how many terms are there in the given arithmetic progression, we have to find out what is the number of the last term, which is given below:
$\Rightarrow {a_n} = 22$
$\Rightarrow a + \left( {n - 1} \right)d = 22$
Substituting the values of $a$ and $d$, in the above equation, which is shown below:
$\Rightarrow 4 + \left( {n - 1} \right)3 = 22$
$\Rightarrow \left( {n - 1} \right)3 = 22 - 4$
On further simplification of the above equation, to get the value of $n$, as given below:
$\Rightarrow \left( {n - 1} \right) = \dfrac{{22 - 4}}{3}$
$\Rightarrow \left( {n - 1} \right) = 6$
$\therefore n = 7$
Hence the last number 22 is the 7th term in the given arithmetic progression.
$\Rightarrow {a_7} = 22$
Thus we are finding the sum of 7 terms in an arithmetic progression, with known initial terms, known common difference and known no. of terms in an arithmetic progression.
The sum of 7 terms of the given arithmetic progression is given by ${S_7}$, is shown below:
$\Rightarrow {S_7} = \dfrac{7}{2}\left[ {2\left( 4 \right) + \left( {7 - 1} \right)\left( 3 \right)} \right]$
$\Rightarrow {S_7} = \dfrac{7}{2}\left[ {8 + 6\left( 3 \right)} \right]$
$\Rightarrow {S_7} = \dfrac{7}{2}\left[ {8 + 18} \right]$
$\Rightarrow {S_7} = \dfrac{7}{2}\left[ {26} \right]$
$\Rightarrow {S_7} = 7 \times 13$
$\therefore {S_7} = 91$

The sum of the arithmetic sequence $4 + 7 + 10 + ..... + 22$ is 91.

Note: Please note that while solving anything in an A.P with an initial term $a$, and with a common difference $d$, then we can find out any term in that particular A.P., where the nth term of the A.P. is given by ${a_n} = a + \left( {n - 1} \right)d$. Here please note that the common difference remains the same throughout the A.P.