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How many terms are there in a geometric series if the first term is 3, the common ratio is 4, and the sum of the series is 1,023?

Respuesta :

In Geometric Progression, we know S(n) = a.(rⁿ - 1 ) / r - 1
Here, a = 3, r = 4 a(n) = 1,023
1023 = 3 (4ⁿ - 1) / (4 - 1)
3(4ⁿ - 1) = 1023 * 3
4ⁿ - 1 = 3069 / 3
4ⁿ = 1023 + 1
4ⁿ = 1024
4⁵ = 1024

In short, Your Answer would be 5 Terms

Hope this helps!



Cricetus

There are five terms. A complete solution is provided below.

Given values are:

First term,

  • a₁ = 3

Common ratio,

  • r = 4

Sum,

  • Sn = 1023

As we know,

→ [tex]S_n = \frac{a_1 (1-r^n)}{1-r}[/tex]

By substituting the given values, we get

→ [tex]1023 = \frac{3(1-4^n)}{1-4}[/tex]

→ [tex]1023= \frac{3(1-4^n)}{3}[/tex]

→  [tex]\frac{3069}{3} = 1-4^n[/tex]

→ [tex]1024 = 4^n[/tex]

→    [tex]4^n = 4^{5}[/tex]

→     [tex]n = 5[/tex]

Thus the above answer is correct.      

Learn more about Geometric series here:

https://brainly.com/question/2617147?referrer=searchResults

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