In an A.P, the first term is 2 and the sum of first five terms is 5, then the ${\text{3}}{{\text{1}}^{st}}$term is:
$
{\text{a}}{\text{. 13}} \\
{\text{b}}{\text{. 17}} \\
{\text{c}}{\text{. - 13}} \\
{\text{d}}{\text{. }}\dfrac{{27}}{2} \\
{\text{e}}{\text{. - }}\dfrac{{27}}{2} \\
$
Last updated date: 23rd Mar 2023
•
Total views: 308.1k
•
Views today: 3.90k
Answer
308.1k+ views
Hint: - ${n^{th}}$term of an A.P is given as$\left( {{a_n} = {a_1} + \left( {n - 1} \right)d} \right)$, (where d is the common difference)
Given data:
First term of an A.P$\left( {{a_1}} \right) = 2$……………… (1)
Sum of first five terms $\left( {{S_5}} \right) = 5$……………… (2)
Then we have to find out the value of ${\text{3}}{{\text{1}}^{st}}$term.
Now, we know that the sum of an A.P is
${{\text{S}}_n} = \dfrac{n}{2}\left( {2{a_1} + \left( {n - 1} \right)d} \right)$, (where d is the common difference)
So, ${{\text{S}}_5} = \dfrac{5}{2}\left( {2{a_1} + \left( {5 - 1} \right)d} \right)$
Now from equation (1) and (2) we have
$
{{\text{S}}_5} = \dfrac{5}{2}\left( {2{a_1} + \left( {5 - 1} \right)d} \right) \\
\Rightarrow 5 = \dfrac{5}{2}\left( {2 \times 2 + 4d} \right) \\
\Rightarrow 2 = 4 + 4d \\
\Rightarrow d = \dfrac{{2 - 4}}{4} = \dfrac{{ - 2}}{4} = \dfrac{{ - 1}}{2} \\
$
Now, we have to find out the value of ${\text{3}}{{\text{1}}^{st}}$term.
As we know that the ${n^{th}}$term of an A.P is given as$\left( {{a_n} = {a_1} + \left( {n - 1} \right)d} \right)$
$ \Rightarrow {31^{th}}$Term of the A.P is
$ \Rightarrow {a_{31}} = 2 + \left( {31 - 1} \right)\left( {\dfrac{{ - 1}}{2}} \right) = \left( {2 - 15} \right) = - 13$
So, option (c) is correct.
Note: - In such types of questions the key concept we have to remember is that always remember all the general formulas of A.P which is stated above, then first find out the value of common difference using the formula of sum of an A.P then using the formula of ${n^{th}}$term of an A.P calculate the value of ${31^{th}}$term which is the required answer.
Given data:
First term of an A.P$\left( {{a_1}} \right) = 2$……………… (1)
Sum of first five terms $\left( {{S_5}} \right) = 5$……………… (2)
Then we have to find out the value of ${\text{3}}{{\text{1}}^{st}}$term.
Now, we know that the sum of an A.P is
${{\text{S}}_n} = \dfrac{n}{2}\left( {2{a_1} + \left( {n - 1} \right)d} \right)$, (where d is the common difference)
So, ${{\text{S}}_5} = \dfrac{5}{2}\left( {2{a_1} + \left( {5 - 1} \right)d} \right)$
Now from equation (1) and (2) we have
$
{{\text{S}}_5} = \dfrac{5}{2}\left( {2{a_1} + \left( {5 - 1} \right)d} \right) \\
\Rightarrow 5 = \dfrac{5}{2}\left( {2 \times 2 + 4d} \right) \\
\Rightarrow 2 = 4 + 4d \\
\Rightarrow d = \dfrac{{2 - 4}}{4} = \dfrac{{ - 2}}{4} = \dfrac{{ - 1}}{2} \\
$
Now, we have to find out the value of ${\text{3}}{{\text{1}}^{st}}$term.
As we know that the ${n^{th}}$term of an A.P is given as$\left( {{a_n} = {a_1} + \left( {n - 1} \right)d} \right)$
$ \Rightarrow {31^{th}}$Term of the A.P is
$ \Rightarrow {a_{31}} = 2 + \left( {31 - 1} \right)\left( {\dfrac{{ - 1}}{2}} \right) = \left( {2 - 15} \right) = - 13$
So, option (c) is correct.
Note: - In such types of questions the key concept we have to remember is that always remember all the general formulas of A.P which is stated above, then first find out the value of common difference using the formula of sum of an A.P then using the formula of ${n^{th}}$term of an A.P calculate the value of ${31^{th}}$term which is the required answer.
Recently Updated Pages
Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts
Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE
