NCERT Class 10 Chapter 3 Arithmetic Progressions Official CBSE Board Sample Problems Long Answer

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Question

Divide 56 in four parts in A.P such that the ratio of the product of their extremes (1st and 4th) to the product of means (2nd and 3rd) is 5: 6.

Solution

Let the four parts of the A.P. are

Now,[ Sum=56]

According to question,

Thus,4 parts are .

Question

The sums of first n terms of three arithmetic progressions are and respectively.

The first term of each AR is 1 and their common differences are 1, 2 and 3 respectively.

Prove that

Solution

Here, sum of n terms of AP is

[where ]

[where]

Here, sum of n terms of AP is

[where ]

[where]

Here, sum of n terms of AP is

[where ]

[where]

Here, sum of n terms of AP is

[where]

[Where]

Here, sum of n terms of AP is

[ Where

[ Where

Now, consider

Question

The digits of a positive number of three digits are in A.P. and their sum is 15. The number obtained by reversing the digits is 594 less than the original number. Find the number.

Solution

Let the required numbers in AR arc respectively.

Now, (Sum of digits)

According to question, number is

, i.e.

Number on reversing the digits is

i.e.

Now, as per given condition in question,

Digits of number arc

Required number is

Question

If the ratio of the sum of first n terms of two A.P.’s is, find the ratio of their mth terms.

Solution

…(i)

Since

So replacing by ,i.e. in (i)

Thus, the ratio of their mth terms is

Question

If the sum of first 7 terms of an A.P. is 49 and that of its first 17 terms is 289, find the sum of first n terms of the A.P

Solution

Given:

, where

… (i)

Similarly,

... (ii)

Solving (1) and (ii), we get

and

Question

The sum of first 9 terms of an AP is 81 and the sum of its first 20 terms is 400. Find the first term and the common difference of the AP.

Solution

Solving the two equations we get

and

Question

513 logs are stacked in the following manner; 54 logs at the bottom row, 51 in the next row, 48 in the row next to it and so on. In how many rows are the 513 logs placed and how many logs are in the top row?

Solution

Number of rows = 18

Number of logs in top row

Question

The sum of four consecutive numbers in an AP is 32 and the ratio of the product of the first and the last term to the product of two middle terms is 7:15. Find the numbers.

Solution

Let the four numbers in AP be a

Numbers are 2, 6, 10, 14(when) and 14, 10, 16 and

Question

If the sum of m terms of an AP is the same as the sum of its n terms, show that the sum of its terms is zero.

Solution

Let first term of A.P. be ‘a’ and common difference be ‘d’.

Sum of terms

Hence sum of terms is zero

Question

How many terms of the AP 24, 21, 18... Must be taken so that the sum is 78? Explain the double answer.

Solution

First term,

Common difference.

Let the number of terms to get sum 78 is n.

Solving the quadratic equation, we get and.

So you can take either 4 terms or 13 terms to get the sum 78.

The reason for the double answer is that the sum of terms from 5th to 13th is zero as some terms are positive and some terms are negative.