Which term of the following sequences:

(a) 2, $$2 \sqrt{2}$$, 4,… is 128

(b) $$\sqrt{3}$$, 3, $$3 \sqrt{3}$$,… is 729

(c) $$\dfrac{1}{3}$$, $$\dfrac{1}{9}$$, $$\dfrac{1}{27}$$, … is $$\dfrac{1}{19683}$$

Asked by Abhisek | 1 year ago |  76

##### Solution :-

(a) The given sequence, 2,$$2 \sqrt{2}$$, 4,…

We have,

a = 2 and r =$$\dfrac{2\sqrt{2}}{2}$$$$\sqrt{2}$$

Taking the nth term of this sequence as 128, we have

$$a_n=ar^{n-1}$$

$$(2)(\sqrt{2})^{n-1}=128$$

$$(2)(2)^\dfrac{n-1}{2}=(2)^7$$

$$(2)^\dfrac{n-1}{2}+1=7$$

$$\dfrac{n-1}{2}=6$$

n-1=12

n = 13

Therefore, the 13th term of the given sequence is 128.

(ii) Given sequence, $$\sqrt{3}$$, 3, $$3\sqrt{3}$$,…

We have,

a = $$\sqrt{3}$$ and r = $$3\sqrt{3}$$$$\sqrt{3}$$

Taking the nth term of this sequence to be 729, we have

$$a_n=ar^{n-1}$$
$$ar^{n-1}=729$$

$$(3)^{\dfrac{1}{2}}$$$$(3)^{\dfrac{n-1}{2}}$$= (3)6

$$(3)^{\dfrac{1}{2}+{\dfrac{n-1}{2}}}$$ =(3)6

Equating the exponents, we have

$$\dfrac{1}{2}+{\dfrac{n-1}{2}}$$ =6

$$\dfrac{1+n-1}{2}$$ = 6

n = 12

(iii) Given sequence, $$\dfrac{1}{3}$$, $$\dfrac{1}{9}$$, $$\dfrac{1}{27}$$, …

a = $$\dfrac{1}{3}$$ and r = (1/9)/(1/3) = $$\dfrac{1}{3}$$

Taking the nth term of this sequence to be $$\dfrac{1}{19683}$$, we have

$$a_n=ar^{n-1}=\dfrac{1}{19683}$$

$$( \dfrac{1}{3}) (\dfrac{1}{3})^{n-1}=\dfrac{1}{19683}$$

$$( \dfrac{1}{3}) ^n=( \dfrac{1}{3}) ^9$$

= 9

Therefore, the 9th term of the given sequence is $$\dfrac{1}{19683}$$.

Answered by Pragya Singh | 1 year ago

### Related Questions

#### Construct a quadratic in x such that A.M. of its roots is A and G.M. is G.

Construct a quadratic in x such that A.M. of its roots is A and G.M. is G.

#### Find the two numbers whose A.M. is 25 and GM is 20.

Find the two numbers whose A.M. is 25 and GM is 20.

#### If a is the G.M. of 2 and 1/4 find a.

If a is the G.M. of 2 and $$\dfrac{1}{4}$$ find a.

#### Find the geometric means of the following pairs of numbers

Find the geometric means of the following pairs of numbers:

(i) 2 and 8

(ii) a3b and ab3

(iii) –8 and –2

Insert 5 geometric means between $$\dfrac{32}{9}$$ and $$\dfrac{81}{2}$$.