Let a_n be a geometric progression such that a_1+a_3=25 and a_2+a_4=50. What is the sum of the first 7 terms of this geometric progression?
If the common difference is d, then
a1 = a1
a2 = a1 + d
a3 = a1 + 2d
a4 = a1 + 3d
a1 + a3 = 25 ---> a1 + ( a1 + 2d ) = 25 ---> 2a1 + 2d = 25
a2 + a4 = 50 ---> ( a1 + d ) + ( a1 + 3d ) = 50 ---> 2a1 + 4d = 50
subtracting: -2d = -25
dividing by -2 d = 12.5
Since a1 + a3 = 25 ---> a1 + ( a1 + 2d ) = 25 ---> 2a1 + 2(12.5) = 25
2a1 + 25 = 25
2a1 = 0
a1 = 0
a7 = a1 + 6d ---> a7 = 0 + 6(12.5) ---> a7 = 75
Sum = n( a1 + a7 ) / 2 = 7( 0 + 75 ) / 2 = 262.5