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In a geometric sequence of real numbers, the sum of the first two terms is 11 and the sum of the first six terms is 77.
The sum of the first four terms is:
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We are given that
a + ar = 11, (1)
= . (2)
Regroup and simplify (2) in this way:
+ + = ,
+ + = ,
= .
Now replace (a + ar) by 11, using (1):
= ,
= ---> = ---> = ---> (factor left side) --->
= .
If we solve it in real numbers, then the solution is unique for : = 2. ( = -3 is not acceptable).
Now, the sum of the first four terms is
= + = + = = 11*(1+2) = 11*3 = 33.
Answer. The sum of the first four terms of the progression is 33.