Numbers and Sequences Ex 2.8-10th Std Maths-Book Back Question And Answer
Question 1.
Find the sum of first n terms of the G.P.
(i) 5, -3, 95,-2725, ……
(ii) 256,64,16,…….
Answer:
(i) 5,-3,95,2755, ….. n terms
(ii) 256,64,16,…….
Answer:
Here a = 256, r = 64256 = 14
Question 2.
Find the sum of first six terms of the G.P. 5,15,45,…
Answer:
Here a = 5, r = 153 = 3, n = 6
Sum of first 6 terms = 1820
Question 3.
Find the first term of the G.P. whose common ratio 5 and whose sum to first 6 terms is 46872.
Answer:
Common ratio (r) = 5
S6 = 46872
The first term of the G.P. is 12.
Question 4.
Find the sum to infinity of (i) 9 + 3 + 1 + ….(ii) 21 + 14 + 283 ……
Answer:
(i) 9 + 3 + 1 + ….
a = 9, r = 39 = 13
Sum of infinity term = a1–r = 91−13
Question 5.
If the first term of an infinite G.P. is 8 and its sum to infinity is 323 then find the common ratio.
Answer:
Here a = 8, S∞ = 323
a1−r = 323
81−r = 323
32 – 32 r = 24 ⇒ 32 r = 8
r = 832 = 14
Common ration = 14
Question 6.
Find the sum to n terms of the series
(i) 0.4 + 0.44 + 0.444 + …… to n terms
Answer:
(ii) 3 + 33 + 333 + ………… to n terms
Answer:
Sn = 3 + 33 + 333 + …. to n terms
= 3[1 + 11 + 111 + …. to n terms]
= 39 [9 + 99 + 999 + …. n terms]
= 13 [(10 – 1) + (100 – 1) + (1000 – 1) + …… n terms]
= 13 [10 + 100 + 1000 + ….. n terms – (1 + 1 + 1 ….. n terms)]
Question 7.
Find the sum of the Geometric series 3 + 6 + 12 + …….. + 1536
Answer:
3 + 6 + 12 …. +1536
a = 3, r = 63 = 2
tn = 1536
∴ Sum of the series is 3069
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 ong letter, find the amount spent on postage when 8th set of letters is mailed.
Answer:
When kumar writes a letter to his friend.Friend writes a letter to another person.
It form a G.P
The G.P is 4, 16, 64,………
Here a = 4, r = 4
The last term is 4 (4)8-1 = 4(4)7
Question 9.
Find the rational form of the number 0.123 .
Answer:
Let x = 0.123¯¯¯¯¯¯¯¯¯¯¯
= 0.123123123….
= 0.123 + 0.000123 + 000000123 + ….
This is an infinite G.P
Question 10.
If Sn = (x + y) + (x2 + xy + y2) + (x3 + x2y + xy2 + y3) + ………… n terms then prove that
Answer:
Sn = (x + y) + (x2 + xy + y2) + (x3 + x2y + xy2 + y3) + …….. n
Multiply by x
x Sn = x(x + y) + x(x2 + xy + y2) + x(x3 + x2y + xy2 + y3) + ……….. n
= x2 + xy + x3 + x2y + xy2 + x4 + x3y + x2y2 + xy3 + …… n terms ……(1)
Multiply by y
ySn = y(x + y) + y(x2 + xy + y2) + y(x3 + x2y + xy2 + y3) + ….. n
= xy + y2 + x2y + xy2 + y3 + x3y + x2y2 + xy3 + y4 + ….. n terms
Subtract (1) and (2)
x Sn – y Sn = x2 + xy + x3 + x2y + xy2 + x4 + x3y + x2y2 + xy3 + …….
– xy + y2 + x2y + xy2 + y3 + x3y + x2y2 + xy3 + y4 + ……
(x – y) Sn = (x2 + x3 + x4 + ……) – (y2 + y3 + y4 + ……)
[ a = x2; r = x and a = y2; r = y, Sn = a(rn−1)r−1]
Hence it is proved.
