(Please show me with full work out)

Question

The first the third and the seventh terms of an increasing arithmetic progression are three consecutive terms of geometrical progression.If the first term of the arithmetic progress is 10 , find the common difference of the arithmetic progression.

Answer 1

d=common difference of the arithmetic progression 

 

r=common ratio of the geometric progression

 

for the arithmetic progression, the terms are:

 

10, 10 + d, 10 + 2d, 10 + 3d, ...  

 

so the 3rd term is 10 + 2d and the 7th term is 10 + 6d

 

the terms of the geometric progression are the 1st, 3rd and 7th terms. These are

 

10, 10+ 2d, 10 + 6d

 

if r = common ratio, any term divided by the term before it equals r

 

( 10 + 2d)/10 =r and (10 + 6d)/(10 + 2d) = r

 

set these equal to each other since they both equal r and cross multiply

 

(10 + 2d)(10 + 2d) = 10(10 + 6d)

 

100 + 40d + 4d² = 100 + 60d

 

4d² - 20d = 0

 

4d(d - 5) = 0

 

discard the solution of d=0

 

d=5

 

to check, the terms of the arithmetic progression are

 

10, 15, 20, 25, 30, 35, 40 ...

 

the terms of the geometric progression are

 

10, 20, 40

 

and 20/10 = 2 and 40/20 = 2

 

Another tutor may very well have a quicker solution