# 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 4th term of an A. P. is zero. Prove that the 25th term of the A. P. is three times its 11th 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 9th 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 3rd and 9th 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.

5th 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.

, 48th 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

## 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 9th 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.

9th 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.

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