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.