Question 788794
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2 + 2&#8730;2 + 4

That's a geometric series with common ratio r = &#8730;2 

Geometric sequence:

S<sub>n</sub> = {{{a[1](1-r^n)/(1-r)}}}

S<sub>n</sub> = {{{a[1](1-r^n)/(1-r)}}} 

30 + 14&#8730;2 = {{{2(1-(sqrt(2))^n)/(1-sqrt(2))}}}

(30 + 14&#8730;2)(1 - &#8730;2) = 2·[1 - (&#8730;2)<sup>n</sup>

30 - 16&#8730;2 - 14·2 = 2 - 2(&#8730;2)<sup>n</sup>

Divide every term by 2

15 - 8&#8730;2 - 14 = 1 - (&#8730;2)<sup>n</sup>

1 - 8&#8730;2 = 1 - (&#8730;2)<sup>n</sup>

-8&#8730;2 = -(&#8730;2)<sup>n</sup>

8&#8730;2 = (&#8730;2)<sup>n</sup>

Write 8 as 2<sup>3</sup> and &#8730;2 as 2<sup>.5</sup>

2<sup>3</sup>2<sup>.5</sup> = (2<sup>.5</sup>)<sup>n</sup>

2<sup>3.5</sup> = 2<sup>.5n</sup>

Equate the powers of 2

3.5 = .5n

{{{3.5/.5}}} = n

7 = n

Answer: It will take 7 terms for the sum to amount to 30 + 14&#8730;2   
Edwin</pre>