# 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.