Question

The fifth term of an exponential sequence is 475 and the second term is 35 fine
The third term
The sixth term
Sum of the first five term

Answer 1

In an exponential sequence, each term can be expressed as:

\[
a_n = a_1 \cdot r^{n-1}
\]

where \( a_n \) is the \( n \)-th term, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.

From the information provided:
- The fifth term (\( a_5 \)) is 475.
- The second term (\( a_2 \)) is 35.

Using the definition of an exponential sequence, we have:

\[
a_5 = a_1 \cdot r^{4} = 475
\]
\[
a_2 = a_1 \cdot r^{1} = 35
\]

Now we can express \( a_1 \) in terms of \( r \):

\[
a_1 = \frac{35}{r}
\]

Next, substitute this expression for \( a_1 \) into the equation for \( a_5 \):

\[
475 = \left(\frac{35}{r}\right) \cdot r^{4}
\]

This simplifies to:

\[
475 = 35r^{3}
\]

Solving for \( r^{3} \):

\[
r^{3} = \frac{475}{35}
\]

Calculating the fraction:

\[
r^{3} = 13.57142857 \quad (\text{approx. } 13.57)
\]

To find \( r \):

\[
r = \sqrt[3]{13.57142857} \approx 2.4027
\]

Now we can calculate \( a_1 \):

\[
a_1 = \frac{35}{r} \approx \frac{35}{2.4027} \approx 14.591
\]

### Finding the third term

The third term (\( a_3 \)) is given by:

\[
a_3 = a_1 \cdot r^{2}
\]

Calculating \( a_3 \):

\[
a_3 \approx 14.591 \cdot (2.4027)^2 \approx 14.591 \cdot 5.7757 \approx 84.36
\]

### Finding the sixth term

The sixth term (\( a_6 \)) can be calculated using:

\[
a_6 = a_1 \cdot r^{5}
\]

Calculating \( a_6 \):

\[
a_6 \approx 14.591 \cdot (2.4027)^5 \approx 14.591 \cdot 61.5081 \approx 897.06
\]

### Sum of the first five terms

The sum of the first five terms (\( S_5 \)) in an exponential sequence can be calculated using the formula:

\[
S_5 = a_1 \frac{r^5 - 1}{r - 1}
\]

Calculating \( S_5 \):

\[
S_5 \approx 14.591 \cdot \frac{(2.4027)^5 - 1}{2.4027 - 1} \approx 14.591 \cdot \frac{61.5081 - 1}{1.4027} \approx 14.591 \cdot \frac{60.5081}{1.4027} \approx 14.591 \cdot 43.1559 \approx 630.75
\]

### Summary of Results

- The third term (\( a_3 \)) is approximately **84.36**.
- The sixth term (\( a_6 \)) is approximately **897.06**.
- The sum of the first five terms (\( S_5 \)) is approximately **630.75**.