Math / Algebra

QuestionFind the partial sum of the geometric sequence given $a=3, r=2$ , $\mathrm{n}=8$ .

Studdy Solution

STEP 1

1. The sequence is geometric, meaning each term after the first is found by multiplying the previous term by a constant, called the common ratio $r$ .
2. The first term of the sequence is $a = 3$ .
3. The common ratio is $r = 2$ .
4. The number of terms to sum is $n = 8$ .

STEP 2

1. Identify the formula for the partial sum of a geometric sequence.
2. Substitute the given values into the formula.
3. Calculate the partial sum.

STEP 3

The formula for the partial sum $S_n$ of the first $n$ terms of a geometric sequence is:
$S_n = a \frac{r^n - 1}{r - 1}$
where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

STEP 4

Substitute the given values $a = 3$ , $r = 2$ , and $n = 8$ into the formula:
$S_8 = 3 \frac{2^8 - 1}{2 - 1}$

STEP 5

Calculate the expression:
First, calculate $2^8$ : $2^8 = 256$
Next, substitute back into the formula: $S_8 = 3 \frac{256 - 1}{1}$ $S_8 = 3 \times 255$
Finally, calculate the product: $S_8 = 765$
The partial sum of the geometric sequence is:
$\boxed{765}$

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QuestionFind the partial sum of the geometric sequence given $a=3, r=2$ , $\mathrm{n}=8$ .

Studdy Solution

STEP 1

1. The sequence is geometric, meaning each term after the first is found by multiplying the previous term by a constant, called the common ratio $r$ .
2. The first term of the sequence is $a = 3$ .
3. The common ratio is $r = 2$ .
4. The number of terms to sum is $n = 8$ .

STEP 2

1. Identify the formula for the partial sum of a geometric sequence.
2. Substitute the given values into the formula.
3. Calculate the partial sum.

STEP 3

The formula for the partial sum $S_n$ of the first $n$ terms of a geometric sequence is:
$S_n = a \frac{r^n - 1}{r - 1}$
where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

STEP 4

Substitute the given values $a = 3$ , $r = 2$ , and $n = 8$ into the formula:
$S_8 = 3 \frac{2^8 - 1}{2 - 1}$

STEP 5

Calculate the expression:
First, calculate $2^8$ : $2^8 = 256$
Next, substitute back into the formula: $S_8 = 3 \frac{256 - 1}{1}$ $S_8 = 3 \times 255$
Finally, calculate the product: $S_8 = 765$
The partial sum of the geometric sequence is:
$\boxed{765}$

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QuestionFind the partial sum of the geometric sequence given a=3,r=2a=3, r=2a=3,r=2, n=8\mathrm{n}=8n=8.

Studdy Solution

STEP 1

1. The sequence is geometric, meaning each term after the first is found by multiplying the previous term by a constant, called the common ratio r r r.2. The first term of the sequence is a=3 a = 3 a=3.3. The common ratio is r=2 r = 2 r=2.4. The number of terms to sum is n=8 n = 8 n=8.

STEP 2

1. Identify the formula for the partial sum of a geometric sequence.2. Substitute the given values into the formula.3. Calculate the partial sum.

STEP 3

The formula for the partial sum Sn S_n Sn​ of the first n n n terms of a geometric sequence is:Sn=arn−1r−1 S_n = a \frac{r^n - 1}{r - 1} Sn​=ar−1rn−1​where a a a is the first term, r r r is the common ratio, and n n n is the number of terms.

STEP 4

Substitute the given values a=3 a = 3 a=3, r=2 r = 2 r=2, and n=8 n = 8 n=8 into the formula:S8=328−12−1 S_8 = 3 \frac{2^8 - 1}{2 - 1} S8​=32−128−1​

STEP 5

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QuestionFind the partial sum of the geometric sequence given $a=3, r=2$ , $\mathrm{n}=8$ .

1. The sequence is geometric, meaning each term after the first is found by multiplying the previous term by a constant, called the common ratio $r$ .
2. The first term of the sequence is $a = 3$ .
3. The common ratio is $r = 2$ .
4. The number of terms to sum is $n = 8$ .

1. Identify the formula for the partial sum of a geometric sequence.
2. Substitute the given values into the formula.
3. Calculate the partial sum.

The formula for the partial sum $S_n$ of the first $n$ terms of a geometric sequence is:
$S_n = a \frac{r^n - 1}{r - 1}$
where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Substitute the given values $a = 3$ , $r = 2$ , and $n = 8$ into the formula:
$S_8 = 3 \frac{2^8 - 1}{2 - 1}$