Question

(a) Find x if the following numbers are in A.P

(1) 4, x, 12

(2) 2, (x − 1), 4

(3) 6 − a, x, 6 + a

(b) Find x if the following numbers are in G.P

(1) 16, x, 25

(2) 5, x + 1, 20

(c) If A, G, H are A.M, G.M, and H.M of the following pair of numbers

(a) 1 and 4

(b) 2 and 8

(c) 9 and 16

Verify that (i) A, G, H are in G.P, (ii) A > G > H.

Answer 1

(a)

(1) We need to find x, where is an A.P.

Here, x is the arithmetic mean of 4 and 12.

(2) We need to find x, where is an A.P.

Here, (x − 1) is the arithmetic mean of 2 and 4.

(3) We need to find x, where is an A.P.

Here, x is the arithmetic mean of .

(b)

(1) We need to find x, where is in G.P.

Here, x is the geometric mean of 16 and 25.

(2) We need to find x, where is in G.P.

Here, (x + 1) is the geometric mean of 5 and 20.

(c)

(a) A, G and H are the A.M., G.M. and H.M. of 1 and 4.

(i) We need to verify that A, G, H is a G.P.

Thus, A, G, H is a G.P.

(ii) We need to verify that A > G > H.

It is seen that A > G > H.

(b) A, G and H are the A.M., G.M. and H.M. of 2 and 8.

(i) We need to verify that A, G, H is a G.P.

Thus, A, G, H is a G.P.

(ii) We need to verify that A > G > H.

It is seen that A > G > H.

(c) A, G and H are the A.M., G.M. and H.M. of 9 and 16.

(i) We need to verify that A, G, H is a G.P.

Thus, A, G, H is a G.P.

(ii) We need to verify that A > G > H.

It is seen that A > G > H.