Find the number of terms in the following G.P. (i) 4, 8, 16,…,8192 ?

Find the number of terms in the following G.P.

(i) 4, 8, 16,…,8192 ?

Solution :

Let nth term be "8192"

tn  = 8192

a = 4, r = 8/4  = 2

 ar n -1  =  8192

4(2)n -1  =  8192

22(2)n -1  =  8192

2 n+1  = 213

n + 1  = 13

n = 12

Hence the 12th term of the above geometric sequence is 8192.

Hence there are 12 terms in the given G.P.

Which of the following sequences are in G.P.? (vi) 120,60,30,18,…

Which of the following sequences are in G.P.?

(vi) 120,60,30,18,…

Solution :

First term (a)  = 120, common ratio

r  = t2/t1   =  60/120 r  = 1/2

r  = t2/t1   =  30/60 r = 1/2

Since the common ratios are same, it is geometric sequence.

Which of the following sequences are in G.P.? (v) 1, −5, 25, −125,…

Which of the following sequences are in G.P.?

(v)  1, −5, 25, −125,…

Solution :

First term (a)  = 1/3, common ratio

r  = t2/t1   =  -5/1 r  = -5

r  = t2/t1   =  25/(-5) r = -5

Since the common ratios are same, it is geometric sequence.

Which of the following sequences are in G.P.? (iii) 0.5, 0.05, 0.005,…

Which of the following sequences are in G.P.?

(iii) 0.5, 0.05, 0.005,…

Solution :

First term (a)  = 0.5, common ratio

r  = t2/t1   =  0.05/0.5 r  = 0.1

r  = t2/t1   =  0.005/0.05 r = 0.1

Since the common ratios are same, it is geometric sequence.

Which of the following sequences are in G.P.? (i) 3, 9, 27, 81,…

Which of the following sequences are in G.P.?

(i) 3, 9, 27, 81,…

Solution :

First term (a)  = 3, common ratio

r  = t2/t1   =  9/3 r = 3

r  = t3/t2   =  27/9 r = 3

r  = t2/t1   =  81/27 r = 3

Since the common ratios are the same, it is a geometric sequence.

Find the middle term(s) of an A.P. 9, 15, 21, 27,…,183.

Find the middle term(s) of an A.P. 9, 15, 21, 27,…,183.

Solution :

In order to find the middle term of the sequence, first we have to know how many terms are in the given sequence.

n = [(l - a)d] + 1

a = 9, d = 15 - 9  ==> 6 and l = 183

n = [(183 - 9)/6] + 1

 n = (174 / 6) + 1

n = 29 + 1  = 30

n = 30(even)

middle terms are (n/2)th term and [(n/2) + 1]th term

15th term  = a + 14d =  9 + 14(6) =  93

16th term  = a + 15d =  9 + 15(6) =  99

Find the next three terms of the following sequence. (iii) 1/4, 2/9, 3/16,…

Find the next three terms of the following sequence.

(iii) 1/4, 2/9, 3/16,…

Solution :

Pattern :

General term  = n [1/(n + 1)]2

n is elements of natural numbers.

4th term  = 4 [1/(4 + 1)]2

  =  4[1/5]2

  =  4/25

5th term  = 5 [1/(5 + 1)]2

  =  5[1/6]2

  =  5/36

6th term  = 6 [1/(6 + 1)]2

  =  6[1/7]2

  =  6/49

Hence the next three terms are 4/25, 5/36, 6/49.

Find the next three terms of the following sequence. (ii) 5, 1,-3,…

Find the next three terms of the following sequence.

(ii) 5, 1,-3,…

Solution :

Pattern :

By subtracting 4 from the 1st term, we get second term. By subtracting 4 from 2nd term, we get 3rd term.

4th term  = -3 - 4  = -7

5th term  = -7 - 4  = -11

6th term  = -11 - 4  = -15

Hence the next three terms are -7, -11, -15.

Find the next three terms of the following sequence. (i) 8, 24, 72, …

Find the next three terms of the following sequence.

(i) 8, 24, 72, …

Solution :

To find the next three, first we have to find out the pattern followed in sequence.

Pattern :

Multiplying the first term by 3, we get the second term.Multiplying the second term by 3, we get the third term.

4th term  = 3 (72)  = 216

5th term  = 216 (3)  = 648

6th term  = 648(3)  = 1944

Hence the next three terms are 216, 648, 1944.