Question

The first and last terms of an AP are 1 and 11. If the sum of its terms is 36, then the number of terms will be
a) 5
b) 6
c) 7
d) 8

Answer 1

Hint: A.P stands for arithmetic progression which is basically a sequence of numbers in which the difference of any two consecutive terms is same throughout. We are given about an A.P whose first and last term are given in addition to it we are given the sum of the terms of the A.P and we are asked to find out the total number of terms in the A.P . This can be achieved by using the formulas listed here.
Formulas used:
In an A.P the $ n $ th term is represented as $ {a_n} $ , the common difference is denoted as $ d $ , the sum up to $ n $ terms is represented as $ {S_n} $ .
The formulas involved in an A.P are
1) $ {a_n} = {a_1} + (n - 1)d $
2) $ d = {a_{n + 1}} - {a_n} $
3) $ {S_k} = \sum\limits_{n = 0}^k {{a_n}} = \dfrac{k}{2} \times ({a_1} + {a_k}) $

Complete step-by-step answer:
We are given that the sum of the A.P is 36
So, we have, $ {a_1} + {a_2} + ... + {a_n} = 36 $ --(1)
In the given A.P we have the first term 1 and last term 11.
Let the A.P has n terms so we have,
 $ {a_1} = 1,{a_n} = 11 $
So, by the sum of A.P formula we have,
 $ {S_k} = \sum\limits_{n = 0}^k {{a_n}} = \dfrac{k}{2} \times ({a_1} + {a_k}) $
 $ \Rightarrow \sum\limits_{n = 0}^{k = n} {{a_n}} = \dfrac{n}{2} \times ({a_1} + {a_n}) $
 $ \Rightarrow {a_1} + {a_2} + ... + {a_n} = \dfrac{n}{2} \times (1 + 11) $ --(2)
Comparing (1) and (2) we can see that,
 $ 36 = \dfrac{n}{2} \times (1 + 11) $
 $ \Rightarrow 36 = \dfrac{n}{2} \times 12 $
 $ \Rightarrow 36 = n \times 6 $
 $ \Rightarrow n = 6 $
Therefore, the number of terms in the given A.P is $ 6 $ .
So, the correct answer is “Option B”.

Note: Always remember the general formulas involved in an A.P this will help you in solving the problem and the most important thing is to understand the given data in the question and to put it in the appropriate place in the formula.