   Question Answers

# In an A.P if the $12^{th}$ term is -13 and the sum of its first four terms is 24. Find the sum of its first ten terms.  Hint: Assume an A.P series with ‘a’ as its first term and the difference between the consecutive terms be ‘d’. Our $12^{th}$ term of the A.P is -13. So, we can write it in mathematical form, i.e, $a+11d=-13$ . We have a summation of the first four terms equal to 24. So, we can also write it in mathematical form, i.e, $4a+6d=24$. Now, we have two equations and two unknown variables. Using these two equations, find the values of ‘a’ and ‘d’. Now, it can be solved further and we can find the summation of the first ten terms.

Complete step-by-step solution -
Let us assume an A.P with ‘a’ as its first term and the difference between the consecutive term be ‘d’.
We have the value of the $12^{th}$ term. So, first of all, we have to find the $12^{th}$ term in terms of the variable ‘a’ and ‘d’.
Putting n=12 in the equation, ${{T}_{n}}=a+(n-1)d$ , we get
${{T}_{12}}=a+(12-1)d$ ……………….(1)
According to the question, we have the $12^{th}$ term of an A.P as -13.
Now, we can write equation (1), as
$a+11d=-13$ ……………(2)
We know that the summation of n terms of an A.P is ${{S}_{n}}=\dfrac{n}{2}\left( {{1}^{st\,term}}+\operatorname{Last}\,term \right)$ .
According to the question, we have the summation of the first four terms which is 24.
${{S}_{n}}=\dfrac{n}{2}\left( {{1}^{st\,term}}+\operatorname{Last}\,term \right)$
$24=\dfrac{4}{2}\left( {{1}^{st\,term}}+\operatorname{Fourth}\,term \right)$ ……………(3)
Now, we have to find the $4^{th}$ term of the A.P.
Putting n=4 in the equation, ${{T}_{n}}=a+(n-1)d$ , we get
${{T}_{4}}=a+(4-1)d=a+3d$ …………….(4)
Now, putting equation (4) in equation (3) and we have ‘a’ as the first term of the A.P.
$24=2(a+a+3d)$
$\Rightarrow 12=2a+3d$ ……………………(5)
Using equation (2) and equation (5), we can find the value of ‘a’ and ‘d’.
According to equation (2), we have
$a+11d=-13$
$\Rightarrow a=-11d-13$ …………..(6)
Putting the value of ‘a’ from equation (6) in equation (5), we get
\begin{align} & 12=2a+3d \\ & \Rightarrow 12=2(-11d-13)+3d \\ \end{align}
\begin{align} & \Rightarrow 12=-22d-26+3d \\ & \Rightarrow 12+26=-22d+3d \\ & \Rightarrow 38=-19d \\ & \Rightarrow -2=d \\ \end{align}
We have got the value of ‘d’ which is -2.
Now, putting the value of ‘d’ in equation (6), we get
\begin{align} & a=-11d-13 \\ & \Rightarrow a=-11(-2)-13 \\ & \Rightarrow a=22-13 \\ & \Rightarrow a=9 \\ \end{align}
The 1st term of the A.P is 9.
Now, we have to find the summation of the first ten terms of the A.P.
Putting n=10 in the equation, ${{S}_{n}}=\dfrac{n}{2}\left( {{1}^{st\,term}}+\operatorname{Last}\,term \right)$ , we get
${{S}_{10}}=\dfrac{10}{2}\left( {{1}^{st\,term}}+\operatorname{Last}\,term \right)$ ………….(7)
Our last term is the $10^{th}$ term of the A.P. So, we have to find the $10^{th}$ term of the A.P.
Putting n=10 in the equation, ${{T}_{n}}=a+(n-1)d$ , we get
${{T}_{10}}=a+(10-1)d$
$\Rightarrow {{T}_{10}}=a+9d$ …………………(8)
Now, putting the values of ‘a’ and ‘d’ in equation (8), we get
${{T}_{10}}=9+9(-2)=-9$ …………..(9)
Now, we have got the $10^{th}$ term of the A.P.
Now, putting equation (9) in equation (7), we get
\begin{align} & {{S}_{10}}=\dfrac{10}{2}\left( {{1}^{st\,term}}+\operatorname{Last}\,term \right) \\ & \Rightarrow {{S}_{10}}=5(9-9)=0 \\ \end{align}
So, the summation of the first ten-term of the A.P is 0.

Note: We can also solve the summation using the formula, ${{S}_{n}}=\dfrac{n}{2}\left[ 2a+(n-1)d \right]$ .
We have, n=10, a=9 and d=-2.
Putting the values of a,n and d in the above formula, we get
${{S}_{10}}=\dfrac{10}{2}\left[ 2(9)+(10-1)(-2) \right]=5\left[ 18-18 \right]=0$ .
Hence, the summation of the first ten terms of the A.P is 0.

View Notes
The Difference Between an Animal that is A Regulator and One that is A Conformer  To Measure the Volume of an Irregular Lamina Using Screw Gauge  What is the full form of phd?  Central Problems of an Economy  An Overview of Food Chain  Significance of Genetics in the Process of Evolution  Potential Energy of Charges in an Electric Field  What Happens if the Earth Stops Rotating?  Energy of An Orbiting Satellite  The Idea of Time  Important Questions for CBSE Class 7 English An Alien Hand Chapter 4 - The Cop and The Anthem  Important Questions for CBSE Class 7 English An Alien Hand Chapter 1 - The Tiny Teacher  Important Questions for CBSE Class 7 English An Alien Hand Chapter 3 - The Desert  Important Questions for CBSE Class 7 English An Alien Hand Chapter 9 - A Tiger In The House  Important Questions for CBSE Class 7 English Honeycomb Chapter 10 - The Story of Cricket  CBSE Class 8 Science Reaching The Age of Adolescence Worksheets  Important Questions for CBSE Class 8 Science Chapter 10 - Reaching The Age of Adolescence  Important Questions for CBSE Class 10 Maths Chapter 13 - Surface Areas and Volumes  Important Questions for CBSE Class 7 English An Alien Hand Chapter 10 - An Alien Hand  Important Questions for CBSE Class 7 English An Alien Hand Chapter 8 - The Bear Story  Previous Year Question Paper of CBSE Class 10 English  CBSE Class 10 Maths Question Paper 2017  CBSE Class 10 Maths Question Paper 2020  Maths Question Paper for CBSE Class 10 - 2011  Maths Question Paper for CBSE Class 10 - 2008  Maths Question Paper for CBSE Class 10 - 2012  Maths Question Paper for CBSE Class 10 - 2009  Maths Question Paper for CBSE Class 10 (2016)  Maths Question Paper for CBSE Class 10 - 2010  Maths Question Paper for CBSE Class 10 - 2007  NCERT Solutions for Class 7 English An Alien Hand - The Cop And The Anthem  RS Aggarwal Solutions Class 10 Chapter 13 - Constructions (Ex 13A) Exercise 13.1  NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas and Volumes in Hindi  NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas and Volumes (Ex 13.4) Exercise 13.4  NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas and Volumes (Ex 13.3) Exercise 13.3  NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas and Volumes (Ex 13.1) Exercise 13.1  NCERT Solutions for Class 7 English An Alien Hand - A Tiger In The House  RD Sharma Class 10 Solutions Chapter 13 - Exercise 13.2  RD Sharma Class 10 Solutions Chapter 13 - Exercise 13.1  NCERT Solutions for Class 10 Social Science India and the Contemporary World - II Chapter 1 - The Rise of Nationalism in Europe  