Question

Find A5 for the geometric series in which S6 = 63 and the common ratio r = 2.

Answer 1

The value of A5 for the given geometric series, where S6 = 63 and the common ratio ( r = 2 ), is 3.

Step-by-step explanation:

In a geometric series, the sum of the first n terms
`(\(S_n\))` is given by the formula
`\(S_n = (A_1(r^n - 1))/(r - 1)\)`, where
`\(A_1\)` is the first term, (r) is the common ratio, and (n) is the number of terms.

Here, we are given that
`\(S_6 = 63\)` and (r = 2). Using this information, we can set up the equation:

`\[63 = (A_1(2^6 - 1))/(2 - 1)\]`

Solving for
`\(A_1\)`, we get
`\(A_1 = 3\).` Now, to find
`\(A_5\),` we use the formula
`\(A_n = A_1 \cdot r^((n-1))\).` Substituting
`\(A_1 = 3\) and \(r = 2\)`, we find:

`\[A_5 = 3 \cdot 2^((5-1))\]`

Calculating this expression, we get
`\(A_5 = 3 \cdot 2^4 = 3 \cdot 16 = 48\).` Therefore, the value of
`\(A_5\)` for the given geometric series is indeed 48.

In summary, by utilizing the formula for the sum of a geometric series and the given information about
`\(S_6\)`and the common ratio (r), we determined the first term
`\(A_1\)` to be 3. Subsequently, applying the formula for the nth term in a geometric series, we found
`\(A_5\)` to be 48.

Answer 2

The fifth term
`$a_5$` in the geometric series is 16.

To find the fifth term
`$a_5$` in a geometric series, you can use the formula:

`S_n=a_1 (\left(r^n-1\right))/((r-1))`

where:

-
`$S_n$` is the sum of the first n terms,

-
`$a_1$` is the first term,

-
`$r$` is the common ratio.

In this case, you are given that
`$S_6=63$` and
`$r=2$`. Plugging in these values, you get:

`63=a_1 (\left(2^6-1\right))/((2-1))`

Simplify the expression on the right side:

`\begin{aligned}& 63=a_1 ((64-1))/(1) \\& 63=a_1 * 63\end{aligned}`

Now, divide both sides by 63 to solve for
`$a_1$` :

`a_1=1`

Now that you have
`$a_1$`, you can find
`$a_5$` using the formula for the n-th term in a geometric series:

`a_n=a_1 * r^((n-1))\\& a_5=1 * 2^4 \\& a_5=1 * 16 \\& a_5=16`

Therefore, the fifth term
`$a_5$` for the given geometric series is 16 .