Sum of first three terms

[Guest]

In a geometric progression, the first term is 153 and the sixth term is 17/22. What is the sum of the first three terms of the sequence.

Answer: 221

[Guest]

I think you got the question wrong.

If the sixth term is [tex]17/22[/tex], the sum of the first three terms is [tex]224.5835\ldots[/tex]

If however the sixth term is [tex]17/27[/tex], the sum of the first three terms is exactly [tex]221[/tex].
The first six terms of a geometric series look like [tex]a, ar, ar^2, ar^3, ar^4, ar^5[/tex] for some [tex]a[/tex] and [tex]r[/tex] you have to determine.
You are told the first term [tex]a[/tex] is [tex]153[/tex].
You are told the sixth term [tex]ar^5[/tex] is [tex]17/27[/tex], we know that [tex]a=153[/tex], so that means
[tex]153r^5 = 17/27[/tex]
which rearranges to
[tex]r^5 = \frac{17}{27\times 153} = \frac{1}{243}[/tex]
[tex]r = \sqrt[5]{1/243} = 1/3[/tex]
So the first three terms are [tex]153, 153/3, 153/9[/tex], which simplify to [tex]153, 51, 17[/tex], and sum to [tex]221[/tex].

Hope this helped,

R. Baber.