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Rd Chapter 9 Arithmetic Progressions Ex 9.6 Solutions

Question - 51 : - Show that the sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667.

Answer - 51 : -


Hence proved.

Question - 52 : - Find the sum of all integers between 84 and 719, which are multiples of 5.

Answer - 52 : -


Question - 53 : - Find the sum of all integers between 50 and 500, which are divisible by 7.

Answer - 53 : -


Question - 54 : - Find the sum of all even integers between 101 and 999.

Answer - 54 : - All integers which are even, between 101 and 999 are = 102, 104, 106, 108, … 998

Question - 55 : -
(i) Find the sum of all integers between 100 and 550, which are divisible by 9.
(ii) all integers between 100 and 550 which are not divisible by 9.
(iii) all integers between 1 and 500 which are multiplies of 2 as well as of 5.
(iv) all integers from 1 to 500 which are multiplies 2 as well as of 5.
(v) all integers from 1 to 500 which are multiplies of 2 or 5.

Answer - 55 : -


= 250 x 251 + 505 x 50– 25 x 510
= 62750 + 25250 – 12750
= 88000 – 12750
= 75250

Question - 56 : - Let there be an A.P. with first term ‘a’, common difference d. If andenotes its nth term and S the sum of first n terms, find.
(i) n and S , if a = 5, d = 3 and an = 50.
(ii) n and a, if an = 4, d = 2 and Sn = -14.
(iii) d, if a = 3, n = 8 and Sn = 192.
(iv) a, if an = 28, Sn = 144 and n = 9.
(v) n and d, if a = 8, an = 62 and Sn = 210.
(vi) n and an, if a = 2, d = 8 and Sn = 90.
(vii) k, if Sn = 3n2 + 5n and ak =164.

Answer - 56 : - In an A.P. a is the first term, d, the common difference a is the nthterm and Sn is the sum of first n terms,

Question - 57 : - If Sn denotes the sum of first n terms of an A.P., provethat S12 = 3(S8 – S4). 

Answer - 57 : -


Question - 58 : - A thief, after committing a theft runs at a uniform speed of 50 m/minute. After 2 minutes, a policeman runs to catch him. He goes 60 m in first minute and increases his speed by 5m/minute every succeeding minute. After how many minutes, the policeman will catch the thief? [CBSE 2016]

Answer - 58 : -

Let total time be 22 minutes.
Total distance covered by thief in 22 minutes = Speed x Time
= 100 x n = 100n metres
Total distance covered by policeman

Question - 59 : - The sums of first n terms of three A.P.S are S1, S2 andS3. The first term of each is 5 and their common differences are 2,4 and 6 respectively. Prove that S1 + S3 = 2S2. 

Answer - 59 : -


Question - 60 : - Resham wanted to save at least 76500 for sending her daughter to school next year (after 12 months). She saved ₹450 in the first month and raised her savings by ₹20 every next month. How much will she be able to save in next 12 months? Will she be able to send her daughter to the school next year?

Answer - 60 : -

Given : Resham saved₹450 in the first month and raised her saving by ₹20 every month and saved innext 12 months.
First term (a) = 450
Common difference (d) = 20
and No. of terms (n) = 12
We know sum of n terms is in A.P.
Sn = (n/2) [2a + (n – 1) d]
Sn = (12/2) [2x 450 + (12 – 1) x 20]
=> Sn = 6[900 + 240]
=> Sn = 6720
Here we can see that Resham saved ₹ 6720 which is more than ₹ 6500.
So, yes Resham shall be able to send her daughter to school.

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