NCERT Class 10 Chapter 3 Arithmetic Progressions Official CBSE Board Sample Problems Short Answer (For CBSE, ICSE, IAS, NET, NRA 2022)

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Question

If the ratio of sum of the first m and n terms of an A. P. is , show that the ratio of its mth and nth terms is

Solution

Let and be the sum of first m and n terms of the A. P. Let first term and common difference of an A. P. is a and d respectively. Then

Consider,

Hence, ratio of m and n ‘t’ term is 2m — 1: 2, , — 1.

Question

The 4^th term of an A. P. is zero. Prove that the 25^th term of the A. P. is three times its 11^th term.

Solution

Let a be first term and d be the common difference of the A. P. Then

[ Given, ]

Now

Hence,

Since

Question

How many terms of the A. P. 27,24, 21, … should be taken so that their sum is zero?

Solution

In the given A. P.

Here, first term (a)

Common difference (d)

Sum of n terms

Therefore,

Thus, the sum of 19 terms of given A. P. is zero.

Question

How many terms of the A. P. 18,16, 14, … be taken so that their sum is zero?

Solution

Let the number of terms taken for sum to be zero he n.

Then, sum of ii terms

First term (a) = 18

Common difference (d)

Therefore,

Hence, sum of 19 terms is 0.

Question

Find the 9^th term from the end (towards the first term) of the A. P. 5, 9,13 181,185.

Solution

Reversing the given A. P. , we get

185,181, 174, … , 9,5

Now, first term (a)

Common difference, (d)

We know that nth term of an A. R is given by

Ninth term

Question

If the 3^rd and 9^th term of an AP are 4 and respectively, which term of the AP is zero?

Solution

Suppose the nth term of the AP is zero.

5^th term is 0

Question

Find the first negative term of the sequence 185, 181,177, 173, …

Solution

Suppose the first negative term is the nth term.

, 48^th term is the first negative term

Question

The sum of the first n terms of an AP is given by . Find the sixteenth term of the AR

Solution

Question

Two AP՚s have the same common difference. The first terms of the AP՚s are 39 and 58 respectively. What is the difference between their terms?

Solution

Let the common difference in the two AP՚s be d. Difference between sixteenth terms is

Question

Find the 9^th term from the end of the sequence 5,9, 13, … 185.

Solution

Consider the reversed sequence 185 13,9, 5, it is an AP, with first difference as 185 and the common difference as 4.

9^th term from the end

Question

If m times the mth term of an A. P. is equal to n times its n ′ term, then show that term of the A. P. is zero.

Solution

The general n th term of an AP is .

From the given conditions,

Rejecting the non-trivial case of m = n, we assume that m and n are different.

Question

Among the natural numbers 1 to 49, find a number x, such that the sum of numbers preceding it is equal to sum of numbers proceeding it.

Solution

1.2. 3.4 … … 49

( is not between 1 to 49 therefore rejected)

Question

Find the sum of all natural numbers between 200 and 400 which are divisible by 7.

Solution

The first term after 200 divisible by 7 is 203. The last term before 400 divisible by 7 is 399.

The number of terms = ?

Hence the number of terms is 29.

Applying Sum formula we get.

Hence the sum of all the natural numbers divisible by 7 lying between 200 and 400 is 8729.