Question

How do you find the sum of the arithmetic sequence \[2 + 5 + 8 + ..... + 56\]?

Answer 1

Hint: Here, we will first find the common difference between the two terms of the given series. Then we will substitute all the values in the formula of the \[{n^{th}}\] term of arithmetic progression to get the number of terms in the sequence. Finally, we will substitute all the values in the formula of the sum of \[{n^{th}}\] terms of an arithmetic sequence to get the desired answer.

Formula used:
We will use the following formulas:
\[{S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)\], where \[n = \] number of terms,\[a = \] First term,\[d = \] Common difference.
 \[{n^{th}}\] term formula: \[a + \left( {n - 1} \right)d = \] Last term, where, \[n = \] number of terms,\[a = \] First term,\[d = \] Common difference.

Complete step by step solution:
The sequence given to us is: \[2 + 5 + 8 + ..... + 56\]
Here, the value of the first term is \[a = 2\].
Now we will find the common difference, \[d\].
\[d = 5 - 2 = 8 - 5 = 3\]
Also, the last term is 56.
Now we have to find the number of terms by using the \[{n^{th}}\] term formula.
Substituting the values of first term, last term and common difference in the formula \[a + \left( {n - 1} \right)d = \] Last term, we get
\[2 + \left( {n - 1} \right) \times 3 = 56\]
Multiplying the terms, we get
\[ \Rightarrow 2 + 3n - 3 = 56\]
Rewriting the equation, we get
\[ \Rightarrow 3n = 56 + 3 - 2\]
Adding and subtracting the like terms, we get
\[ \Rightarrow n = \dfrac{{57}}{3} = 19\]
Now we will find the sum of the given sequence.
Substitute \[a = 2,d = 3\] and \[n = 19\] in the formula \[{S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)\], we get
\[{S_{19}} = \dfrac{{19}}{2}\left( {2 + \left( {19 - 1} \right) \times 3} \right)\]
Simplifying the expression, we get
\[ \Rightarrow {S_{19}} = \dfrac{{19}}{2}\left( {2 + 54} \right)\]
\[ \Rightarrow {S_{19}} = 532\]

Therefore, the sum of the arithmetic sequence is 532.

Note:
We can find the sum of the arithmetic sequence by adding the first and last term and then divide the sum by two.
Now we know that,
First term \[ = 2\]
Last term \[ = 56\]
Common difference \[ = 5 - 2 = 3\]
\[{n^{th}}\] term formula: \[a + \left( {n - 1} \right)d = \] Last term
\[2 + \left( {n - 1} \right) \times 3 = 56\]
Multiplying the terms, we get
\[ \Rightarrow 2 + 3n - 3 = 56\]
Rewriting the equation, we get
\[ \Rightarrow 3n = 56 + 3 - 2\]
Adding and subtracting the like terms, we get
\[ \Rightarrow n = \dfrac{{57}}{3} = 19\]
Now average of first and last term \[ = \dfrac{{2 + 56}}{2} = \dfrac{{58}}{2} = 29\]
Now we will find the sum of the sequence by multiplying the average by the number of terms in the sequence.
Sum of sequence \[ = \] Number of terms \[ \times \] Average of first and last term
Substituting the values, we get
\[ \Rightarrow \] Sum of Sequence \[ = 19 \times 29 = 532\]