Problem 23 - Calculus for Biology and Medicine

Question

Find the next four values of the sequence \(\left\\{a_{n}\right\\}\) on the basis of the values of \(a_{0}, a_{1}, a_{2}, a_{3}, a_{4}\). $$ \sqrt{1+e}, \sqrt{2+e^{2}}, \sqrt{3+e^{3}}, \sqrt{4+e^{4}}, \sqrt{5+e^{5}} $$

Step-by-Step Solution

Answer

The next four values are \( \sqrt{6+e^6}, \sqrt{7+e^7}, \sqrt{8+e^8}, \sqrt{9+e^9} \).

Step 1: Identify the Sequence Pattern

Observe that the sequence is given in the form of \( \sqrt{n+e^n} \) for \( n = 1, 2, 3, 4, 5 \). This means the general term \( a_n \) can be written as \( \sqrt{n+1+e^{n+1}} \).

Step 2: Determine the Next Value

To find the next value in the sequence, substitute \( n = 5 \) into the general term to get \( a_5 = \sqrt{6 + e^6} \).

Step 3: Calculate the 6th Term

Compute the 6th term of the sequence using the formula: \( a_6 = \sqrt{6 + e^6} \).

Step 4: Determine the Seventh Value

Continue the pattern by substituting \( n = 6 \) to find the 7th term: \( a_6 = \sqrt{7 + e^7} \).

Step 5: Calculate the 7th Term

Compute the 7th term of the sequence: \( a_7 = \sqrt{7 + e^7} \).

Step 6: Determine the Eighth Value

Substitute \( n = 7 \) to find the 8th term: \( a_8 = \sqrt{8 + e^8} \).

Step 7: Calculate the 8th Term

Compute the 8th term of the sequence: \( a_8 = \sqrt{8 + e^8} \).

Step 8: Determine the Ninth Value

Substitute \( n = 8 \) to find the 9th term: \( a_9 = \sqrt{9 + e^9} \).

Step 9: Calculate the 9th Term

Compute the 9th term of the sequence: \( a_9 = \sqrt{9 + e^9} \).

Key Concepts

Understanding Sequence Pattern

In mathematical sequences, identifying a pattern is the key to predicting future terms. A sequence is a list of numbers in a specific order, where each number follows a distinct rule or pattern. To illustrate and expand your understanding of sequence patterns, consider the sequence provided in the exercise:

• \( \sqrt{1+e} \)
• \( \sqrt{2+e^2} \)
• \( \sqrt{3+e^3} \)
• \( \sqrt{4+e^4} \)
• \( \sqrt{5+e^5} \)

By observing these terms, one can ascertain that each term is structured as \( \sqrt{n+e^n} \), starting from \( n = 1 \). This recognizable pattern is crucial to predicting the subsequent terms without guesswork. The pattern is a systematic way to establish relationships between terms using a particular formula. Look carefully for any repetitions or evolving systems that establish how the terms change successively.

Calculating Subsequent Terms

Once the pattern is identified, the next step in working with sequences is term calculation. This involves substituting the appropriate "next" value into the identified pattern or formula. For our specific sequence, once you've identified the pattern \( \sqrt{n+e^n} \), calculating the next terms follows these straightforward steps:

• Determine the position \( n \) for the term you need, for example, \( n = 5, 6, 7, \) and so forth.
• Plug \( n \) into the general pattern. So, for the term where \( n = 5 \), the calculation is \( a_5 = \sqrt{6 + e^6} \).
• Repeat the substitution as needed for additional terms like \( n = 6 \) or \( n = 7 \).

The calculated terms involve basic substitution and following the established pattern. This method not only lends itself to accuracy but also ensures that you're correctly adhering to the sequence's intended formulation.

Deriving the General Term

The beauty of sequences lies in being able to express any term succinctly using a general term. A general term of a sequence provides a direct formula through which you can find any desired term by simply plugging in the term's position number. In this exercise, our general term is derived as:

• For the given sequence, the recognizable structure is \( a_n = \sqrt{n+e^n} \).
• Hence, to find any term \( a_n \), you substitute the value of \( n \) into \( \sqrt{n+e^n} \), giving you the desired result.

Having this general term at your disposal is exceedingly useful because it simplifies the process of finding any term in large sequences without listing all prior terms. Simply put, it streamlines calculations, allowing instant access to any term 'position' within the sequence.