Find S5 for a geometric series for which a1 = 81 and r = 1/9.
To find the S5 (sum of the first 5 terms) for a geometric series, you can use the formula:
S5 = a1 * (1 - r^5) / (1 - r)
In this case, a1 = 81 and r = 1/9.
Plugging in these values into the formula, we have:
S5 = 81 * (1 - (1/9)^5) / (1 - 1/9)
First, let's calculate (1/9)^5:
(1/9)^5 = 1/9 * 1/9 * 1/9 * 1/9 * 1/9 = 1/59049
Now, let's substitute this result back into the formula:
S5 = 81 * (1 - 1/59049) / (1 - 1/9)
Next, simplify the expression:
S5 = 81 * (59049/59049 - 1/59049) / (8/9)
S5 = 81 * (59048/59049) / (8/9)
S5 = (81 * 59048 * 9) / (59049 * 8)
S5 = 603729 / 472392
Finally, divide the numerator by the denominator:
S5 ≈ 1.28
Therefore, the sum of the first 5 terms (S5) for the given geometric series is approximately 1.28.
a r^(n-1)
9^2 (1/9)^4
1/81