Problem 59 - College Algebra

Question

Follow these steps to evaluate a sequence defined recursively using a graphing calculator: • On the home screen, key in the value for the initial term \(a_{1}\) and press [ENTER]. • Enter the recursive formula by keying in all numerical values given in the formula, along with the key strokes \([2 \mathrm{ND}]\) ANS for the previous term \(a_{n-1}.\) Press \([\mathrm{ENTER}].\) • Continue pressing \([\text { ENTER }]\) to calculate the values for each successive term. Use the steps above to find the indicated term or terms for the sequence. Find the first five terms of the sequence \(a_{1}=2, a_{n}=2^{\left[\left(a_{n-1}\right)-1\right]}+1.\)

Step-by-Step Solution

Answer

The first five terms are 2, 3, 5, 17, and 65,537.

Step 1: Identify Initial Term

The initial term of the sequence is given as \(a_1 = 2\). This will be the starting point for our calculation.

Step 2: Input Initial Term

Using your graphing calculator, key in the initial term, which is \(2\). Press [ENTER] to confirm the input.

Step 3: Enter the Recursive Formula

Enter the recursive formula into the calculator. The recursive formula is \(a_n = 2^{[(a_{n-1})-1]} + 1\). Use [2ND] ANS to input \(a_{n-1}\) as it represents the previous term.

Step 4: Calculate Second Term

Press [ENTER] to calculate \(a_2\). With \(a_1 = 2\), the formula becomes:\[a_2 = 2^{(2 - 1)} + 1 = 2^1 + 1 = 3\].Store \(3\) as the second term.

Step 5: Calculate Third Term

Press [ENTER] again to calculate \(a_3\). Use \(a_2 = 3\) in the recursive formula:\[a_3 = 2^{(3 - 1)} + 1 = 2^2 + 1 = 5\].Store \(5\) as the third term.

Step 6: Calculate Fourth Term

Continue by pressing [ENTER] to calculate \(a_4\).With \(a_3 = 5\), compute:\[a_4 = 2^{(5 - 1)} + 1 = 2^4 + 1 = 17\].Store \(17\) as the fourth term.

Step 7: Calculate Fifth Term

Press [ENTER] once more to find \(a_5\).Using \(a_4 = 17\):\[a_5 = 2^{(17 - 1)} + 1 = 2^{16} + 1 = 65,537\].Store \(65,537\) as the fifth term.

Step 8: Conclusion

The first five terms of the sequence are \(2, 3, 5, 17,\) and \(65,537\).

Key Concepts

Graphing Calculator

A graphing calculator is a powerful tool that can evaluate sequences, including those defined recursively. For this task, the graphing calculator helps by allowing users to easily input and manipulate mathematical formulas. When dealing with recursive sequences, the calculator assists in quickly obtaining subsequent terms by repeatedly applying the recursive rule. The calculator's interface lets you input initial conditions and formulas directly, minimizing human error.

• Start by entering the initial term.
• Use functions like "[2ND] ANS" to recall previous terms, aiding in sequence calculations.
• Press '[ENTER]' repeatedly to auto-calculate each term seamlessly.

The calculator streamlines these repetitive calculations, making it significantly easier to handle complex sequences.

Sequence Evaluation

Sequence evaluation involves calculating the individual terms of a sequence. In the context of recursive sequences, each term is dependent on the prior term(s). This makes the process systematic, as each term builds upon the previous one.

• Begin with the known initial term.
• Use the recursive formula to find subsequent terms.
• Continue this process for as many terms as needed.

This method emphasizes the importance of recognizing patterns and understanding the formula's role in generating the sequence.

Initial Term

The initial term of a sequence is critical; it serves as the starting point for evaluating the sequence. In our exercise, this term is denoted as \( a_1 = 2 \).

• It is the first value to be input into the graphing calculator.
• This term propagates through the recursive formula to determine all subsequent terms.

Without a defined initial term, calculating the rest of the sequence becomes impossible, as recursive sequences build directly from this beginning point.

Recursive Formula

A recursive formula for a sequence provides instructions for deriving the next term from the current term or previous terms. In this exercise, the formula is denoted as\[ a_n = 2^{[(a_{n-1})-1]} + 1 \]It requires iterative usage:

• Use the previous term \(a_{n-1}\) to compute the next term \(a_n\).
• Repeated application yields the desired series of terms.

This formula's dynamic nature is what allows it to expand the sequence iteratively, underpinning each term upon its predecessor.