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___
Correct Answer: 2
Solution and Explanation
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 \( 2 \).
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