find the sum of the first 25 terms of a geometric sequence where the first term id 177,147 and the common ratio is -1/3?

First term, a₁ = 177,147
Common ratio, r = -1/3, where |r| < 1

(-1/3)²⁵ ≈ -1.18 × 10⁻¹² ≪ 1

Hence, sum of the first 25 terms, S(25)
≈ Sum to the infinity term, S(∞)
= a₁ / [1 - r]
= 177,147 / [1 - (-1/3)]
= 177,147 / [1 + (1/3)]
= 177,147 / [4/3]
= 177,147 × [3/4]
= 132,860.25

Geometric sequence has the general form of:

a(n) = a₁b^(n - 1)

In this case, a₁ = 177147 and b = -(1/3), so:

a(n) = 177147(-1/3)^(n - 1)

The sum of the first "n" terms of a geometric sequence can be found with this equation:

S(n) = a₁(1 - b^n) / (1 - b)

So plug in the a₁ and b into the above equation, and then solve for S(25):

S(n) = 177147(1 - (-1/3)^n) / (1 - (-1/3))
S(n) = 177147(1 - (-1/3)^n) / (1 + 1/3)
S(n) = 177147(1 - (-1/3)^n) / (4/3)

Division of fractions is multiplication of reciprocal:

S(n) = 177147(1 - (-1/3)^n) * (3/4)
S(n) = 531441(1 - (-1/3)^n) / 4

Now that I've simplified it, solve for S(25):

S(25) = 531441(1 - (-1/3)^25) / 4
S(25) = 531441(1 - (-1/847288609443)) / 4
S(25) = 531441(1 + 1/847288609443) / 4
S(25) = 531441(847288609444/847288609443) / 4
S(25) = 531441(211822152361/847288609443)
S(25) = 112570976472882201 / 847288609443
S(25) = 211822152361 / 1594323

or approximately equal to:

S(25) ≈ 132860.25 (rounded to 2DP)

I also verified this by using Excel.