Problem 9 Let P1, P2, P3... be a sequence defined recursively as follows. [Pr P²=Pk-1+2-3k k>2 an integer P₁ = 2 Use Mathematical Induction to show that for an integer n ≥ 1 P₁ = 3+1_7 Vn>1
Problem 9 Let P1, P2, P3... be a sequence defined recursively as follows. [Pr P²=Pk-1+2-3k k>2 an integer P₁ = 2 Use Mathematical Induction to show that for an integer n ≥ 1 P₁ = 3+1_7 Vn>1
Advanced Engineering Mathematics
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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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![**Problem 9**: Let \( p_1, p_2, p_3, \ldots \) be a sequence defined recursively as follows.
\[
\begin{cases}
p_k = p_{k-1} + 2 \cdot 3^k & \text{for } k > 2 \text{ an integer} \\
p_1 = 2
\end{cases}
\]
Use Mathematical Induction to show that for an integer \( n \geq 1 \)
\[
p_n = 3^{n+1} - 7 \quad \forall \, n \geq 1
\]
**Explanation:**
- The sequence \( (p_n) \) is defined recursively where each term \( p_k \) for \( k > 2 \) is calculated using the formula \( p_k = p_{k-1} + 2 \cdot 3^k \).
- The initial condition for this sequence is \( p_1 = 2 \).
- The goal is to prove, using mathematical induction, that for every integer \( n \geq 1 \), the formula \( p_n = 3^{n+1} - 7 \) holds.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe40c0608-13e4-4b93-bf1d-ab7d88be51a9%2Fa3ed34e7-29d2-4bee-8c2b-5ce7dd698e87%2Fv82a3o_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 9**: Let \( p_1, p_2, p_3, \ldots \) be a sequence defined recursively as follows.
\[
\begin{cases}
p_k = p_{k-1} + 2 \cdot 3^k & \text{for } k > 2 \text{ an integer} \\
p_1 = 2
\end{cases}
\]
Use Mathematical Induction to show that for an integer \( n \geq 1 \)
\[
p_n = 3^{n+1} - 7 \quad \forall \, n \geq 1
\]
**Explanation:**
- The sequence \( (p_n) \) is defined recursively where each term \( p_k \) for \( k > 2 \) is calculated using the formula \( p_k = p_{k-1} + 2 \cdot 3^k \).
- The initial condition for this sequence is \( p_1 = 2 \).
- The goal is to prove, using mathematical induction, that for every integer \( n \geq 1 \), the formula \( p_n = 3^{n+1} - 7 \) holds.
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