Question

Which term of \[37,32,27\ \ldots\] is \[- 103\] ?
A. \[30\]
B. \[29\]
C. \[28\]
D. \[27\]

Answer 1

Hint:In this question, we need to find which term of the given series \[37,32,27\ \ldots\] is \[- 103\] . On observing the given sequence, it is an arithmetic progression. It is nothing but a sequence where the difference between the consecutive terms are the same. From the given series, we can find the first term \[(a)\] and the common difference \[(d)\] . Then , by using the formula of arithmetic progression, we can easily find \[n\]. Here an is the term for which we need to find the value of \[n\], which is given as \[- 103\].

Formula used :
The Formula used to find the nth terms in arithmetic progression is
\[a_{n} = \ a\ + \left( n – 1 \right)d\]
Where \[a\] is the first term
\[d\] is the common difference
\[n\] is the number of term
\[a_{n}\] is the \[n^{\text{th}}\] term

Complete answer:
Given , \[37,32,27,\ldots\]
Here we need to find which term of the given series \[37,32,27\ \ldots\] is \[- 103\] .
Here \[a = 37\] and \[d = (32 – 37)\] which is equal to \[- 5\] .
The formula used to find the \[n^{\text{th}}\] terms in arithmetic progression is \[a_{n} = \ a + \left( n – 1 \right)d\]
Given that \[a_{n}\] is \[- 103\] .
Now on substituting the known values,
We get,
\[\Rightarrow \ - 103 = 37 + (n – 1)( - 5)\]
On simplifying,
We get,
\[- 103 = 37 + ( - 5n + 5)\]
On removing the parentheses,
We get,
\[- 103 = 37 – 5n + 5\]
Now on simplifying,
We get,
\[5n = 103 + 37 + 5\]
On further simplifying,
We get,
\[5n = 145\]
On dividing both sides by \[5\] ,
We get,
\[\Rightarrow \ n = \dfrac{145}{5}\]
On simplifying,
We get,
\[\Rightarrow \ n = 29\]
Thus , \[29^{\text{th}}\] term in the AP series is \[- 103\] .
Final answer :
\[29^{\text{th}}\] term in series \[37,32,27\ \ldots\] is \[- 103\] .
Option B).\[29\] is the correct answer.

Therefore, the correct option is B

Note: In order to solve these types of questions, we should have a strong grip over arithmetic series. We should be very careful in choosing the correct formula because there is the chance of making mistakes in interchanging the formula of finding term and summation of term. If we try to solve this sum with the formula \[S_{n} = \dfrac{n}{2}\left( a + l \right)\ \] where \[l\] is the last term of the series , then our answer will be totally different and can get confused.