Question

The sum of the first 13 consecutive odd numbers is________.
(A). 196
(B). 169
(C). 13
(D). 81

Answer 1

Hint: we can treat this question as an AP and find its sum just by applying one formula or we can also add each and every term starting from 1 till the \[{13^{th}}\] term appears.

Complete step by step answer:
 Let us do it by considering an AP
For that let us write some odd numbers from the starting
1, 3, 5, 7, 9, ………….
So here the first term which is denoted by a and the common difference denoted by d and naturally n will denote the \[{n^{th}}\]
Now we know that the sum of terms of an AP is denoted by \[{S_n} = \dfrac{n}{2}\left\{ {2a + \left( {n - 1} \right)d} \right\}\]
Using this in the above mention AP
It can be observed that
\[\begin{array}{l}
a = 1\\
d = 2\\
n = 13
\end{array}\]
As we are told to find the sum till \[{13^{th}}\]
Let us put all this value in the above mentioned formula and we will get it as
\[\begin{array}{l}
\therefore {S_n} = \dfrac{n}{2}\left\{ {2a + \left( {n - 1} \right)d} \right\}\\
 = \dfrac{{13}}{2}\{ 2a + 12d\} \\
 = \dfrac{{13}}{2} \times 2(a + 6d)\\
 = 13(1 + 6 \times 2)\\
 = 13 \times 13\\
 = 169
\end{array}\]
Therefore option B is the correct option.

Note: We can also do this question just by adding the first 13 odd numbers i.e., \[1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 = 169\] Hence in both the cases we will get the same result.