10th- Maths- Chapter 2- Numbers and Sequences -Exercise 2.8 (All Sums)
Question 1.
Find the sum of first n terms of the G.P.
(i) 5, -3, 95, – 2725, ……….
(ii) 256, 64, 16, ……..
Solution:
(i) 5, -3, 95, −2725, ………
(ii) 256, 64, 16,…….
a = 256
Question 2.
Find the sum of first six terms of the G.P. 5, 15, 45, …
Solution:
G.P.= 5, 15, 45, 15
Question 3.
Find the first term of the GP. whose common ratio 5 and whose sum to first 6 terms is 46872.
Solution:
Common ratio, r = 5
S6 = 46872
Question 4.
Find the sum to infinity of (i) 9 + 3 + 1 + ….
(ii) 21 + 14 + 28/3 + ….
Solution:
(i) 9 + 3 + 1 + ….
(ii) 21 + 14 + 28/3 + ….
Question 5.
If the first term of an infinite G.P. is 8 and its sum to infinity is 32/3 then find the common ratio.
Solution:
a = 8
Question 6.
Find the sum to n terms of the series
(i) 0.4 + 0.44 + 0.444 +…. to n terms
(ii) 3 + 33 + 333 + …. to n terms
Solution:
(i) 0.4 + 0.44 + 0.444 + … to n terms
= 4 (0.1 + 0.11 + 0.111 + … to n terms)
= 4/9 (0.9 + 0.99 + 0.999 + … to n terms)
= 4/9 (1 -0.1) + (1 – 0.01) +(1 – 0.001) + … n terms)
(ii) 3 + 33 + 333 + … to n terms.
= 3 (1 + 11 + 111 + …….. to n terms)
= 3/9 (9 + 99 + 999 +… to n terms)
1/3 = [(10 – 1) + (100 – 1) + (1000 – 1) + … to n terms]
= 1/3 [(10 + 100 + 1000 + .. ) + (-1)n]
Question 7.
Find the sum of the Geometric series
3 + 6 + 12 + …… + 1536.
Solution:
3 + 6 + 12 + …… + 1536
Here a = 3
Question 8.
Kumar writes a letter to four of his friends. He asks each one of them to copy the letter and mail to four different persons with the instruction that they continue the process similarly. Assuming that the process is unaltered and it costs ₹2 to mail one letter, find the amount spent on postage when 8th set of letters is mailed.
Amount spent
[a + ar + ar2 +… arn-1]
= 4 [2 + 2 × 4 + 2 × 42 ÷ + 2 × 47]
= 4 [Sn] Here n = 8, r = 4
Solution:
Kumar (1) 2 × 4
Cost of posting 1st set of Letters.
Cost of posting 2nd set of letters
Amount spent
[a + ar + ar2 +… arn-1]
= 4 [2 + 2 × 4 + 2 × 42 ÷ + 2 × 47]
= 4 [Sn] Here n = 8, r = 4
It is a G.P
∴ Cost of postage after posting 8th set of letters
= 4 × 43690 = ₹ 174760
Question 9.
Find the rational form of the number 0.123¯¯¯¯¯¯¯¯
Solution:
Let x = 0.123123123 ………. ⇒ x = 0.123¯¯¯¯¯¯¯¯ …..(1)
Multiplying 1000 on both rides
1000 x = 123.123123… ⇒ 1000x = 123.123¯¯¯¯¯¯¯¯ …..(2)
(2) – (1) = 1000x – x 123.123¯¯¯¯¯¯¯¯ – 0.123¯¯¯¯¯¯¯¯.
⇒ 999 x = 123
⇒ x = 123/999
⇒ x = 41/333 Rational number
Question 10.
If Sn = (x + y)+(x2 + xy + y2) + (x3 + x2y + xy2 + y3) + …n terms then prove that (x – y)
Solution:
Sn = (x + y) + (x2 + xy + y2) + (x3 + x2y + xy2 + y3) + …n terms
⇒ x. Sn = (x + y)x + (x2 + xy + y2)x + (x3 + x2y + xy2 + y3) x + ….
⇒ x. Sn = x2 +xy + x3 + x2y + y2x + x4 + x3y
+ x2 y2 + y3 x +…… ….(1)
Multiplying ‘y’ on both sides,
= y.Sn = (x + y)y + (x2 + xy + y2) y+
(x3 + x2y + xy2 + y3) y + ….
= y.Sn = xy + y2 + x2y + xy2 + y3 + x3y
+ x3y + x2y2 + xy3 + y4 + ……….
(1) – (2) ⇒
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