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:

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

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Math Problems

Algebra 2

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. $S_6 = 63$