Class 10 Maths Important Questions for Arithmetic Progression
Given below are the Class 10 Maths Important Questions & extra questions for Arithmetic Progression.This exercise has variety of quetsions including tough and difficult question on AP also
a. Concepts questions
b. Calculation problems
c. Long answer questions
d. proof questions
Question 1 The general term of a sequence is given by an = -4n + 15. Is the sequence an A. P.? If so, find its 15th term and the common difference. Solution
$a_n = -4n + 15$
$a_k = -4k + 15$
$a_{k+1} = -4(k+1) + 15$
Now
$a_{k+1} - a_k=-4(k+1) + 15 -[-4k + 15]=-4$
Since difference between two terms constant.It is a AP
$a_{15} = -4(15) + 15=-45$
Question 2 The nth term of an A. P. is 6n + 11. Find the common difference. Solution
Question 3 If the 8th term of an A. P. is 31 and the 15th term is 16 more than the 11th term, find the A. P. Solution
a8 = 31
a +(8-1)d = 31
a + 7d = 31 -- (i)
a15 =16 + a11
a + 14d = 16 + a +10d
14d = 16 + 10d
4d = 16
d = 4
Now putting d = 4 in eq. (i) we get
a + 7(4) = 31
a + 28 = 31
a = 3
So AP
3,7,11,15...
Question 4 Which term of the arithmetic progression 5, 15, 25, ----- will be 130 more than its 31st term? Solution
Let n th term be 130 more than the 31st term of the A.P.
First term of A.P. = 5
Common difference = 15 - 5 = 10
an = 130 + a31
5 + (n - 1) X 10 = 130 + 5 + (31 - 1) X 10
10 (n - 1) = 430
n = 44
Thus, 44th term of the A.P is 130 more than the 31st term.
Question 5 Which term of the A. P. 3, 15, 27, 39...... will be 132 more than its 54th term? Solution
Similar question as above.
Answer is 65th term is 132 more than its 54th term
Question 6 Two A. P.’s has the same common difference. The difference between their 100th terms is 111222333. What is the difference between their Millionth terms? Solution
$a_n=a + (n-1)d$
For Ist AP
$a_{100x}=a_x + (100-1)d$
For 2nd AP
$a_{100y}=a_y + (100-1)d$
So difference
$a_x-a_y=a_{100x}- a_{100y}$
$a_x-a_y=111222333$
This difference will remain in all the terms
So ,answer is 111222333 only
Question 7 The 10th and 18th terms of an A. P. are 41 and 73 respectively. Find 26th term. Solution
Question 11 The sum of 4th and 8th terms of an A. P. is 24 and the sum of 6th and 10th terms is 34. Find the first term and the common difference of the A. P. Solution
$a + 3d + a + 7d=24$ or $2a + 10d=24$
$a+ 5d+ a + 9d=34$ or $2a + 14d =34$
Solving these
a=-1/2 ,d=5/2
Question 12 If an A. P. consists of n terms with first term a and nth term l show that the sum of the nth term from the beginning and the mth term from the end is (a + l). Question 13 If the ath term of an A. P. be 1/b and bth term be 1/a then show that its (ab)th term is 1. Question 14 If the pth term of an A. P. is q and the qth term is p. Prove that its nth term is (p + q – n) Question 15 If m times the mth term of an A. P. is equal to n times its nth term. Show that the (m + n)th term of the A. P. is zero. Solution
According to the question
$m(a_m) = n(a_n)$
Now for a,d as first term and common difference of the AP,nth term is defined as ,
$a_n= a + (n-1)d$
So
$m[a + (m-1)d] = n[a + (n-1)d]$
$[ma + (m^2 - m)d]= [na + (n^2 - n)d]$
$[ma + (m^2d- md)]= [na + (n^2d- nd)]$
$[ma-na] + [m^2d- n^2d] + [nd-md] = 0$
$a[m-n] + d[m^2 - n^2] + d[n-m] = 0$
Now we know that $x^2 - y^2 = (x+y) (x-y)$
$a[m-n] + d[(m+n) (m-n)] - d[m - n] = 0$
Divide the above equation with (m-n) ,We get :
$a + d(m+n) - d = 0$
$a + [ (m+n) - 1 ] d = 0$
So $a_{m+n} = 0$
Question 16 Justify whether it is true to say that the following are the nth terms of an AP.
(i) $2n -3$
(ii) $3n^2+5$
(iii) $1+n+n^2$ Solution
We can solve these question in three ways
a. Find the Ist,2nd ,3rd and 4th term and check for the common difference. If same then in AP
b. Find d as $d=a_n - a_{n-1}$. Now if d does not depend on n, then it is in AP
c. We know that in an AP $a_n=a + (n-1)d$. We observe that $a_n$ is a linear polynomial in n. So if the expression is not linear,it is not an AP
i. $a_n=2n -3$
$a_{n-1} =2(n-1) -3=2n-5$
$d=a_n - a_{n-1}= 2n -3 - 2n + 5= 2$
So this is an nth term of A.P
ii. $a_n =3n^2+5$
This is not an linear polynomial,so it is not an nth term of A.P
iii. $a_n=1+n+n^2$
This is not an linear polynomial,so it is not an nth term of A.P
Question 17
Find the sum of 3 digit numbers which are not divisible by 7? Solution
Sum of all three digit numbers which are not divisible by 7
= Sum of all three digit numbers - Sum of all three digit numbers which are divisible by 7
Now,lets find each of these separately
Sum of all three digit numbers
= 100 + 101 + 102 +...... + 999
Now this series is a AP with first term as 100, common difference =1 and n=900
$S_1 = \frac {n}{2} [a_1 + a_n] = \frac {900}{2} [100 + 999]= 494500$
Sum of all three digit numbers which are divisible by 7
= 105 + 112 + ..... + 994
Now this series is a AP with first term as 105, common difference=7, last term =994, n=?
Now from nth term formula
$a_n= a_1 + (n-1) \times d$
$994 = 105 + (n - 1) \times 7$
$7(n - 1) = 889$
$n = 128$
So, sum of all three digit numbers which are divisible by 7
$S_2 = \frac {n}{2} [a_1 + a_n] = \frac {128}{2} [105 + 994]= 70336$
Therefore,
Sum of all three digit numbers which are not divisible by 7 = $S_1 - S_2 = 494500 - 70336 = 424164$
Question 18
In an AP, if $S_n = 3n^2+ 5n$ and ak = 164, find the value of k. Solution
Given: $S_n = 3n^2+ 5n$
$S_1 = 3(1)^2 + 5(1)
= 3 + 5=8$
Therefore
$a_1 = 8$
$S_2 = 3(2)^2 + 5(2)
= 12 + 10
= 22
$
Therefore
$a_1+ a_2 =22$
$
8 + a_2 = 22$
$a_2 = 14$
So, the AP is
8 ,22......
So, common difference=14
Now from nth term formula
$a_n= a_1 + (n-1) \times d$
$164 = 8 + (k-1)14$
$k = 27
$
Question 19
If $S_n$ denotes the sum of first n terms of an AP, prove that
$S_{12}= 3(S_8- S_4)$ Solution
Let a is the first term of A.P and d is the common difference
$
S_n=\frac {n}{2} {2a+(n-1)d}$
$S_{12}=\frac {12}{2} {2a+(12-1) d}=12a+66d
$
$_S8=\frac {8}{2} {2a+7d}=8a+28d$
$
S_4=\frac {4}{2} {2a+3d}=4a+6d$
Question 21
Find the 20th term from the end of the AP 3, 8, 13......253 Question 22
The sum of the first n terms of an A.P whose first term is 8 and the common difference is 20 is equal to the sum of first 2n terms of another AP whose first term is - 30 and the common difference is 8. Find n. Solution
This Class 10 Maths Important Questions for Arithmetic Progression with answers is prepared keeping in mind the latest syllabus of CBSE . This has been designed in a way to improve the academic performance of the students. If you find mistakes , please do provide the feedback on the mail.
link to this page by copying the following text Also Read
