Find the first two terms of a geometric sequence if t_3+ t_4 = 50 and t_4 + t_5 =100?

Question

t3= a(r^2)
t4= a(r^3)
t5= a(r^4)

a(r^2)+ a(r^4)= 100
-------------------------
a(r^3)+ a(r^4) = 50

ar^3
----- = 2
ar^2

r = 2 <-------

Answer 1

a(r^3)+ a(r^4)= 100
------------------------
a(r^2)+ a(r^3) = 50

ar^4
----- = 2
ar^2

ar^2 = 2

r = 2^(1/2)

Answer 2

ar^2 + ar^3 = 50
ar^2(1+r) = 50 ---- #1

ar^3 + ar^4 = 100
ar^3(1+r) = 100 --- #2

divide #2 by #1

ar^3/ar^2 = 2
r = 2

in #1:
a(4)(3) = 50
12a = 50
a = 50/12 = 25/6

so the first 5 terms are
25/6 , 50/6, 100/6, 200/6, 400/6

check"
3rd + 4th = 50
LS = 100/6 + 200/6
= 300/6 = 50 , yeah

4th + 5th = 100
LS = 200/6 + 400/6 = 100 , yeah

Answer 3

Thank you