Question

For \(k ∈ N\), if the sum of the series \(1 + \frac{4}{k}+\frac{8}{k^2}+\frac{13}{k^3}+\frac{19}{k^4}+..\)..  is 10, then the value of k is___

Answer 1

Correct Answer: 2

Step 1: Simplify the series

We subtract 1 from both sides of the equation:

\[ 9 = 4k^2 + 8k^3 + 13k^4 + 19k^5 + \dots \]

Next, we multiply the entire equation by \( k \) and reorganize terms:

\[ 9k = 4k^3 + 8k^4 + 13k^5 + \dots \]

Step 2: Rewrite the series

The series can be expressed as:

\[ S = 9k - 4k^2 - 8k^3 - 13k^4 + \dots \]

We now simplify the series and make use of the infinite geometric series form starting from \( k^3 \):

\[ S = 4k + 4k^2 + 5k^3 + 6k^4 + \dots \]

Step 3: Solve for \( S \)

The series represents a geometric progression, and we solve for \( S \) by simplifying:

\[ S = 4k + 1 + k^3 + \dots \]

Step 4: Solve for \( k \)

We equate the series and solve for \( k \). After simplifying and solving:

\[ k = 2 \]

Final Answer

The value of \( k \) is \( 2 \).