Question

In a geometric progression, the sum of the 2nd term and the 3rd term is 36. The sum of the 3rd term and 4th term is 12. Find the 1st term and the common ratio.​

Answer 1

Answer: a₀ = 81, r = 1/3

Explanation:

a, b, c, d, ....

b + c = 36 → b = 36 - c

c + d = 12 → d = 12 - c

`\text{Geometric progression has the following proportion:}\\.\qquad \qquad \qquad (c)/(b)=(d)/(c)`

Use substitution method to solve for c:

`(c)/(36-c)=(12-c)/(c)\\\\\text{Cross Multiply:}\\c(c)=(36-c)(12-c)\\c^2=432-48c+c2\\0=432-48c\\48c=432\\c=9`

Plug in c = 9 to find the common ratio (r):

`r=(c)/(36-c)\quad =(9)/(36-9)\quad =(9)/(27)\quad =(1)/(3)`

Find the first term (a₀) using a₃ = 9:

`a_n=a_o\bigg((1)/(3)\bigg)^(n-1)\\\\\\a_3=a_o\bigg((1)/(3)\bigg)^(3-1)\\\\\\9=a_o\bigg((1)/(3)\bigg)^(2)\\\\\\9=a_o\bigg((1)/(9)\bigg)\\\\\\81=a_o`