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Since the geometric series has 6 terms, increases by a factor of 4, and has a sum of 1365, the value of the first term is 1.

What is the sum of a geometric series?

The sum of a geometric series is given by

Sₙ = a(rⁿ - 1)/(r - 1) with r > 1 where

  • n = number of terms,
  • a = first term and
  • r = common ratio

Now, since our Geometric Series has 6 terms, n = 6. Also, it increases by a factor of 4, so, r = 4 and has a sum of 1365, so Sₙ = 1356. So,we have that

  • n = 6,
  • Sₙ = S₆ = 1365 and
  • r = 4

The value of the first term

Since we require the first term, a , making a subject of the formula, we have

a = Sₙ(r - 1)/(rⁿ - 1)

Substituting the values of the variables into the equation, we have

a = Sₙ(r - 1)/(rⁿ - 1)

a = S₆(r - 1)/(r⁶ - 1)

a = 1365(4 - 1)/(4⁶ - 1)

a = 1365(3)/(4096 - 1)

a = 4095/4095

a = 1

So, the value of the first term is 1.

Learn more about geometric series here:

https://brainly.com/question/2899779

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