The first three terms of a geometric series are $5, 15, 45$ $5, 15, 45$ . If the nth term of this series is $10935$ $10935$ , what is the sum of the first n terms?

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The first three terms of a geometric series are $5, 15, 45$ $5, 15, 45$ . If the nth term of this series is $10935$ $10935$ , what is the sum of the first n terms?

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Answer 1 · 2017-03-30T01:03:50

16400

Explanation:

This is a geometric series with r = 3. First, find what term $10935$ $10935$ is.

$5 \cdot (3^{n - 1}) = 10935$ $5 * (3^(n-1)) = 10935$

$(3^{n - 1}) = 2187$ $(3^(n-1)) = 2187$

$3^{n - 1} = 3^{7}$ $3^(n-1) = 3^7$

$n - 1 = 7$ $n-1 = 7$

$n = 8$ $n = 8$

Now use the formula for the sum of the first $n$ $n$ terms of a geometric series:

$S_{n} = \frac{a_{1} (1 - r^{n})}{1 - r}$ $S_n = (a_1(1 - r^n))/(1-r)$

$S_{8} = \frac{5 (1 - 3^{8})}{1 - 3} = 16400$ $S_8 = (5(1 - 3^8)) / (1 - 3) = 16400$

Final Answer

Answer 2 · 2017-03-30T01:13:46

The sum is $16, 400$ $16,400$ .

Explanation:

Step 1: Classify the sequence

Since $t_{2} = 3 t_{1}$ $t_2 = 3t_1$ and $t_{3} = 3 t_{2}$ $t_3 = 3t_2$ , this sequence is geometric with $r = 3$ $r = 3$ .

Step 2: Find the number of terms

There is no formula we can use to evaluate the sum without knowing the number of terms. By the formula $t_{n} = a {(r)}^{n - 1}$ $t_n = a(r)^(n - 1)$ , we have:

$10935 = 5 {(3)}^{n - 1}$ $10935 = 5(3)^(n - 1)$

$2187 = 3^{n - 1}$ $2187 = 3^(n - 1)$

$3^{7} = 3^{n - 1}$ $3^7 = 3^(n - 1)$

$7 = n - 1$ $7 = n - 1$

$n = 8$ $n = 8$

Step 3: Evaluate the sum

The formula for the sum of a geometric series is $s_{n} = \frac{a (1 - r^{n})}{1 - r}$ $s_n = (a(1 - r^n))/(1 - r)$ .

$s_{8} = \frac{5 (1 - 3^{8})}{1 - 3}$ $s_8 = (5(1 - 3^8))/(1 - 3)$

$s_{8} = \frac{- 32800}{- 2}$ $s_8 = (-32800)/(-2)$

$s_{8} = 16, 400$ $s_8 = 16,400$

Practice Exercises

$1$ $1$ . Find the sum:

$2 + 8 + 32 +128 + ... + 524,288$

Solution

$1.$

$699,050$

Hopefully this helps!

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The first three terms of a geometric series are $5, 15, 45$ $5, 15, 45$ . If the nth term of this series is $10935$ $10935$ , what is the sum of the first n terms?

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Explanation:

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The first three terms of a geometric series are $5, 15, 45$ $5, 15, 45$ . If the nth term of this series is $10935$ $10935$ , what is the sum of the first n terms?