Question

The first,third and seventh term of an arithmetic progression form three consecutive terms of a geometric progression. If the first, second terms of the arithmetic progression is 6. Find its first term and common difference.

Answer 1

just use your usual formulas, and if the two sequences are
a, ar, ar^2, ...
b, b+d, B=2d, ...
then you have
a = b
ar^2 = b+d
ar^6 = b+2d
so now you have

ar^2 = a+d
ar^6 = a+2d
Now, I don't know what "the first, second terms of the arithmetic progression is 6" is supposed to mean, but I guess it means either of
the first term of the arithmetic progression is 6
the sum of first, second terms of the arithmetic progression is 6
So either
ar^2 = a+6
ar^6 = a+12
r^4 + r^2 + 1 = 2
no integer solutions there..
or,
ar^2 = a+d
ar^6 = a+2d
2a+d = 6
again, no easy solutions.

So I suspect some kind of typo.

Answer 2

If your sentence means:

If the first of the arithmetic progression is 6.

then

In A.P.

an = a1 + ( n - 1 ) d

where

a1 = the initial term

d = common difference

an = the nth term

a3 = a1 + ( 3 - 1 ) ∙ d = a1 + 2 d

a7 = a1 + ( 7 - 1 ) ∙ d = a1 + 6 d

The members of G.P. are:

a1 , a3 , a7

Common ratio of G.P. is the quotient of two adjacent members of G.P.

r = a3 / a1 = ( a1 + 2 d ) / a1

r = a7 / a3 = ( a1 + 6 d ) / ( a1 + 2 d )

r = r

( a1 + 2 d ) / a = ( a1 + 6 d ) / ( a1 + 2 d )

Cross multiply.

( a1 + 2 d )² = a1 ∙ ( a1 + 6 d )

a1² + 2 ∙ a1 ∙ 2 d + ( 2 d )² = a1² + 6 a1 ∙ d

a1² + 4 a1 ∙ d + 4 d² = a1² + 6 a1 ∙ d

Subtract a1² to both sides.

4 a1 ∙ d + 4 d² = 6 a1 ∙ d

Subtract 4 a1 ∙ d to both sides.

4 d² = 2 a1 ∙ d

Divide both sides by 4

d² = 2 a1 ∙ d / 4

d² = a1 ∙ d / 2

Divide both sides by d

d = a1 / 2

Since a1 = 6

d = 6 / 2 = 3

a3 = a1 + 2 d

a3 = 6 + 2 ∙ 3 = 6 + 6 = 12

a7 = a1 + 6 d

a7 = 6 + 6 ∙ 3 = 6 + 18 = 24

Common ratio of this G.P is:

r = a3 / a1 = 12 / 6 = 2

Or, which is the same thing:

r = a7 / a3 = 24 / 12 = 2

A.P:

6 , 6 + 3 , 6 + 2 ∙ 3 , 6 + 3 ∙ 3 , 6 + 4 ∙ 3 , 6 + 5 ∙ 3 , 6 + 6 ∙ 3

6 , 9 , 12 , 15 , 18 , 21 , 24

first term = 6

third term = 12

seventh term = 24

G.P:

6 , 12 , 24

Answer 3

nicely done, @Bosnian. I guess I'll have to take a second look at my analysis.