Suppose , and for , . Find the values of . Can you find a simpler recursive formula for ? Can you give an explicit formula?

Understand the Question We are given a sequence where each term for is defined as the sum of all preceding terms plus , with initial values and . By calculating the first few values ( through ), we can observe how each term relates to the immediately preceding term to find a simpler recurrence relation. Recognizing the resulting pattern of powers of allows us to write a closed-form explicit formula for . Step 1 · Calculate the Values of to Given and . Using for Step 2 · Find a Simpler Recursive Formula For , express by grouping the terms up to Since , substitute this into the expression Also note that . Hence, the simpler recursive relation is Step 3 · Find the Explicit Formula Express each term as a power of Therefore, the explicit formula is for all integers . Answer > Values: Recursive formula: for (with ) Explicit formula: for Common Mistakes Index Off-by-One: Writing the explicit formula as instead of . For , . Missing the Constant Term: Forgetting the in the original summation formula when calculating the first few terms by hand. Base Case Domain: Forgetting to specify the base condition when stating the recursive formula .

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Question 14

Suppose $P_1 = 1$ , $P_2 = 2$ and for $n > 2$ , $P_n = P_1 + P_2 + \dots + P_{n-1} + 1$ . Find the values of $P_1, P_2, \dots, P_8$ . Can you find a simpler recursive formula for $P_n$ ? Can you give an explicit formula?

Solution

Understand the Question

We are given a sequence where each term for $n > 2$ is defined as the sum of all preceding terms plus $1$ , with initial values $P_1 = 1$ and $P_2 = 2$ .
By calculating the first few values ( $P_1$ through $P_8$ ), we can observe how each term relates to the immediately preceding term to find a simpler recurrence relation.
Recognizing the resulting pattern of powers of $2$ allows us to write a closed-form explicit formula for $P_n$ .

Step 1 · Calculate the Values of $P_1$ to $P_8$

Given $P_1 = 1$ and $P_2 = 2$ .

Using $P_n = P_1 + P_2 + \dots + P_{n-1} + 1$ for $n > 2$

\begin{aligned} P_3 &= P_1 + P_2 + 1 = 1 + 2 + 1 = 4 \\ P_4 &= P_1 + P_2 + P_3 + 1 = 1 + 2 + 4 + 1 = 8 \\ P_5 &= P_1 + P_2 + P_3 + P_4 + 1 = 1 + 2 + 4 + 8 + 1 = 16 \\ P_6 &= 1 + 2 + 4 + 8 + 16 + 1 = 32 \\ P_7 &= 1 + 2 + 4 + 8 + 16 + 32 + 1 = 64 \\ P_8 &= 1 + 2 + 4 + 8 + 16 + 32 + 64 + 1 = 128 \end{aligned}

Step 2 · Find a Simpler Recursive Formula

For $n > 2$ , express $P_n$ by grouping the terms up to $P_{n-2}$

P_n = (P_1 + P_2 + \dots + P_{n-2} + 1) + P_{n-1}

Since $P_{n-1} = P_1 + P_2 + \dots + P_{n-2} + 1$ , substitute this into the expression

P_n = P_{n-1} + P_{n-1} = 2P_{n-1}

Also note that $P_2 = 2 = 2P_1$ . Hence, the simpler recursive relation is

P_n = 2P_{n-1} \quad \text{for } n \ge 2, \text{ with } P_1 = 1

Step 3 · Find the Explicit Formula

Express each term as a power of $2$

\begin{aligned} P_1 &= 1 = 2^0 \\ P_2 &= 2 = 2^1 \\ P_3 &= 4 = 2^2 \\ P_4 &= 8 = 2^3 \\ &\;\;\vdots \\ P_n &= 2^{n-1} \end{aligned}

Therefore, the explicit formula is $P_n = 2^{n-1}$ for all integers $n \ge 1$ .

Answer

Values: $P_1 = 1, P_2 = 2, P_3 = 4, P_4 = 8, P_5 = 16, P_6 = 32, P_7 = 64, P_8 = 128$

Recursive formula: $P_n = 2P_{n-1}$ for $n \ge 2$ (with $P_1 = 1$ )

Explicit formula: $P_n = 2^{n-1}$ for $n \ge 1$

Common Mistakes

Index Off-by-One: Writing the explicit formula as $P_n = 2^n$ instead of $P_n = 2^{n-1}$ . For $n = 1$ , $P_1 = 2^{1-1} = 2^0 = 1$ .
Missing the Constant Term: Forgetting the $+1$ in the original summation formula when calculating the first few terms by hand.
Base Case Domain: Forgetting to specify the base condition $P_1 = 1$ when stating the recursive formula $P_n = 2P_{n-1}$ .

Question 14

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