Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Find the sum of the first 6 terms of the following sequence. Round to the nearest hundredth if necessary.

32,quad16,quad8,dots
Sum of a finite geometric series:

S_(n)=(a_(1)-a_(1)r^(n))/(1-r)
Answer:

Find the sum of the first $6$ terms of the following sequence. Round to the nearest hundredth if necessary. $\newline$ $32, \quad 16, \quad 8, \ldots$ $\newline$ Sum of a finite geometric series: $\newline$ $S_{n}=\frac{a_{1}-a_{1} r^{n}}{1-r}$ $\newline$ Answer:

View step-by-step help

View step-by-step help

Sum of finite series not start from 1

Full solution

Q. Find the sum of the first

6

terms of the following sequence. Round to the nearest hundredth if necessary.

\newline

32, \quad 16, \quad 8, \ldots

\newline

Sum of a finite geometric series:

\newline

S_{n}=\frac{a_{1}-a_{1} r^{n}}{1-r}

\newline

Answer:

Identify Terms and Ratio: Identify the first term $a_1$ and the common ratio $r$ of the geometric sequence. $\newline$ The first term $a_1$ is $32$ . $\newline$ The sequence is halving each time, so the common ratio $r$ is $\frac{1}{2}$ .
Use Sum Formula: Use the formula for the sum of the first $n$ terms of a geometric series: $S_n = \frac{a_1 - a_1 \cdot r^n}{1 - r}$ . Here, $n = 6$ , $a_1 = 32$ , and $r = \frac{1}{2}$ .
Substitute and Calculate: Substitute the values into the formula and calculate the sum. $\newline$ $S_6 = \frac{32 - 32 \times (\frac{1}{2})^6}{1 - \frac{1}{2}}$
Calculate $(\frac{1}{2})^6$ : Calculate the value of $(\frac{1}{2})^6.$ $\newline$ $(\frac{1}{2})^6 = \frac{1}{64}$
Substitute Value: Substitute the value of $(\frac{1}{2})^6$ into the formula. $S_6 = \frac{32 - 32 \times \frac{1}{64}}{1 - \frac{1}{2}}$
Simplify Expression: Simplify the expression. $\newline$ $S_6 = \frac{32 - \frac{32}{64}}{\frac{1}{2}}$ $\newline$ $S_6 = \frac{32 - \frac{1}{2}}{\frac{1}{2}}$
Perform Subtraction and Division: Simplify further by performing the subtraction and division. $\newline$ $S_6 = \frac{31.5}{\frac{1}{2}}$ $\newline$ $S_6 = 31.5 \times 2$
Calculate Final Value: Calculate the final value. $S_6 = 63$

More problems from Sum of finite series not start from 1

Question

Find the sum of the finite arithmetic series.

\sum_{n=1}^{10} (7n+4)

\newline

Posted 3 months ago

Question

Type the missing number in this sequence: `55,\59,\63,\text{_____},\71,\75,\79`

Posted 2 months ago

Question

Type the missing number in this sequence: `2, 4 8`, ______,`32`

Posted 2 months ago

Question

What kind of sequence is this?

2, 10, 50, 250, \ldots

Choices:Choices:

\newline

\text{[A]arithmetic}

\newline

\text{[B]geometric}

\newline

\text{[C]both}

\newline

\text{[D]neither}

Posted 2 months ago

Question

What is the missing number in this pattern?

1, 4, 9, 16, 25, 36, 49, 64, 81, \_\_\_\_

Posted 2 months ago

Question

Classify the series.

\sum_{n = 0}^{12} (n + 2)^3

\newline

\newline

\text{[A]arithmetic}

\newline

\text{[B]geometric}

\newline

\text{[C]both}

\newline

\text{[D]neither}

Posted 2 months ago

Question

Find the first three partial sums of the series.

\newline

1 + 6 + 11 + 16 + 21 + 26 + \cdots

\newline

Write your answers as integers or fractions in simplest form.

\newline

S_1 =

\newline

S_2 =

\newline

S_3 =

Posted 2 months ago

Question

Find the third partial sum of the series.

\newline

3 + 9 + 15 + 21 + 27 + 33 + \cdots

\newline

Write your answer as an integer or a fraction in simplest form.

\newline

S_3 =

Posted 2 months ago

Question

Find the first three partial sums of the series.

\newline

1 + 7 + 13 + 19 + 25 + 31 + \cdots

\newline

Write your answers as integers or fractions in simplest form.

\newline

S_1 =

\newline

S_2 =

\newline

S_3 =

Posted 3 months ago

Question

Does the infinite geometric series converge or diverge?

\newline

1 + \frac{3}{4} + \frac{9}{16} + \frac{27}{64} + \cdots

\newline

\newline

\text{[A]converge}

\newline

\text{[B]diverge}

Posted 2 months ago