Mixing arithmetic and geometric progressions
Algebra
Level
3
Consider an increasing arithmetic progression, whose first term is 1 and 2nd , 10th and 34th are in a geometric progression.
Find the common difference of this arithmetic progression.
The answer is 0.33.
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as 2nd 10th and 34th term of the A.P are in G.P
{a+(2-1)d} {a+(34-1)d}=[a+(10-1)d]^2
and we know that, a=1
so,
(1+d).(1+33d)=(1+9d)^2
1+d+33d+33d^2 = 1+81d^2 +18d
48d^2 - 16d=0
d(48d-16)=0
d=0,
d=16/48=1/3, d=1/3
But the question says that common difference is positive
So , d=0.33