Question

There
are 5 geometric means
between a and b. If the second
and last mean are
2 and 16
respectively, find the values of
a and b?​

Answer 1

Answer:

a=8,b=19

Step-by-step explanation:

please mark me as branliest pleazzzzzzzzz. and follow me.

Answer 2

Answer:

a = $\frac{1}{2}$

b = 32

Step-by-step explanation:

Let $a_1,a_2,a_3,a_4,a_5$ be the five geometric mean between 'a' and 'b',

Then we have,

$a,a_1,a_2,a_3,a_4,a_5, b$ form a GP with first term 'a' and last term(7th term) is b

Let 'r' be the common ratio of this GP.

We know,

the nth term of a GP  = $ar^{n-1}$, where 'a' is the first term and 'r' is the common ratio

Given,

Second mean = $a_2$ = 2

Last mean = $a_5$ = 16

Second mean = third term of the GP

that is, $ar^{2}$ = 2 ------------- (1)

Last mean = 6th term of the GP

that is $ar^{5}$ = 16 ------------ (2)

Divide (2) by (1) we get,

$\frac{ar^{5} }{ar^2} = \frac{16}{2}$

$r^{3}$ = 8

r = 2

Substitute the value of 'r' in equation (1) we get,

a x 4 = 2

a = $\frac{2}{4 }= \frac{1}{2}$

Again, b is the 7th term of the GP. Then

b = $ar^(7-1) = ar^6$

Substitue the value of 'a' and 'r'

b = $\frac{1}{2} * 2^{6} = 2^{5} = 32$

Hence,

a = $\frac{1}{2}$ and b = 32