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
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 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.