Chapter 9 Sequences and Series Ex 9.3 Solutions
Question - 1 : - Find the 20th and nth terms of the G.P. 5/2, 5/4, 5/8, ………
Answer - 1 : -
Given G.P.is 5/2, 5/4, 5/8, ………
Here, a =First term = 5/2
r = Common ratio = (5/4)/(5/2) = ½
Thus, the 20th termand nth term
Question - 2 : - Find the 12th term of a G.P.whose 8th term is 192 and the common ratio is 2.
Answer - 2 : -
Given,
The common ratio ofthe G.P., r = 2
And, let a bethe first term of the G.P.
Now,
a8 = ar 8–1 = ar7
ar7 = 192
a(2)7 = 192
a(2)7 = (2)6 (3)
Question - 3 : - The 5th, 8th and 11th terms ofa G.P. are p, q and s, respectively.Show that q2 = ps.
Answer - 3 : -
Let’s take a tobe the first term and r to be the common ratio of the G.P.
Then according to thequestion, we have
a5 = a r5–1 = a r4 = p …(i)
a8 = a r8–1 = a r7 = q …(ii)
a11 = a r11–1 = a r10 = s …(iii)
Dividing equation (ii)by (i), we get
Question - 4 : - The 4th term of a G.P. is square of its second term, and the first term is –3. Determine its 7th term.
Answer - 4 : -
Let’s consider a tobe the first term and r to be the common ratio of the G.P.
Given, a =–3
And we know that,
an = arn–1
So, a4 = ar3 =(–3) r3
a2 = a r1 = (–3) r
Then from thequestion, we have
(–3) r3 =[(–3) r]2
⇒ –3r3 =9 r2
⇒ r =–3
a7 = a r 7–1 = a r6 =(–3) (–3)6 = – (3)7 = –2187
Therefore, the seventhterm of the G.P. is –2187.
Question - 5 : - Which term of the following sequences:
(a) 2, 2√2, 4,… is 128 ? (b) √3, 3, 3√3,… is 729 ?
(c) 1/3, 1/9, 1/27, … is 1/19683 ?
Answer - 5 : -
(a) The givensequence, 2, 2√2, 4,…
We have,
a = 2 and r = 2√2/2 =√2
Taking the nth termof this sequence as 128, we have
Therefore, the 13th termof the given sequence is 128.
(ii) Given sequence,√3, 3, 3√3,…
We have,
a = √3 and r = 3/√3 =√3
Taking the nth termof this sequence to be 729, we have
Therefore, the 12th termof the given sequence is 729.
(iii) Given sequence,1/3, 1/9, 1/27, …
a = 1/3 and r =(1/9)/(1/3) = 1/3
Taking the nth termof this sequence to be 1/19683, we have
Therefore, the 9th termof the given sequence is 1/19683.
Question - 6 : - For what values of x, the numbers -2/7, x, -7/2 arein G.P?
Answer - 6 : -
The given numbers are-2/7, x, -7/2.
Common ratio =x/(-2/7) = -7x/2
Also, common ratio= (-7/2)/x = -7/2x
Therefore, for x =± 1, the given numbers will be in G.P.
Question - 7 : - Find the sum to 20 terms in the geometric progression 0.15, 0.015, 0.0015…
Answer - 7 : -
Given G.P., 0.15,0.015, 0.00015, …
Here, a =0.15 and r = 0.015/0.15 = 0.1
Question - 8 : - Find the sum to n terms in the geometric progression √7,√21, 3√7, ….
Answer - 8 : -
The given G.P is √7,√21, 3√7, ….
Here,
a = √7 and
Question - 9 : - Find the sum to n terms inthe geometric progression 1, -a, a2, -a3 …. (if a ≠-1)
Answer - 9 : -
The given G.P.is 1, -a, a2, -a3 ….
Here, the first term= a1 = 1
And the common ratio= r = – a
We know that,
Question - 10 : - Find the sum to n terms inthe geometric progression x3, x5, x7, … (if x≠ ±1 )
Answer - 10 : -
Given G.P. is x3,x5, x7, …
Here, we have a = x3 and r = x5/x3 =x2
