Problem 13 - Advanced Calculus

Question

Find the first six terms of the sequence defined by \(a_{n}=(-2)^{n+1}+(-3)^{n}\)

Step-by-Step Solution

Answer

The first six terms of the sequence are -5, 1, -43, 49, -307, and 601.

Step 1: Calculate the first term

For the first term, substitute \(n = 1\) into the sequence formula. Hence, \(a_{1}=(-2)^{1+1}+(-3)^{1} = -2 -3 = -5\).

Step 2: Calculate the second term

For the second term, substitute \(n=2\) into the sequence formula. \(a_{2}=(-2)^{2+1}+(-3)^{2}= -8+9=1\).

Step 3: Calculate the third term

For the third term, substitute \(n = 3\) into the sequence formula. Hence, \(a_{3}=(-2)^{3+1}+(-3)^{3} = -16 -27 = -43\).

Step 4: Calculate the fourth term

For the fourth term, substitute \(n = 4\) into the sequence formula. Hence, \(a_{4}=(-2)^{4+1}+(-3)^{4} = -32 + 81 = 49\).

Step 5: Calculate the fifth term

For the fifth term, substitute \(n = 5\) into the sequence formula. Hence, \(a_{5}=(-2)^{5+1}+(-3)^{5} = -64 -243 = -307\).

Step 6: Calculate the sixth term

For the sixth term, substitute \(n = 6\) into the sequence formula. Hence, \(a_{6}=(-2)^{6+1}+(-3)^{6} = -128 + 729 = 601\).

Key Concepts

Term Calculation

In sequences, finding the specific terms involves substituting the term number into the given formula. For example, with the sequence \(a_n = (-2)^{n+1} + (-3)^n\), each unique term is found by replacing \(n\) with the desired term number.

You follow these steps:

• Substitute \(n\) into the formula: For each term, you pick a value for \(n\), starting from 1 upwards for the terms you wish to calculate.


• Simplify the expression: After substituting, follow the rules of arithmetic and exponents to find the term value.

Some of the key calculations include keeping track of negative signs and ensuring correct order of operations.

With practice, term calculation becomes more intuitive, and you gain a better grasp of how each formula is affecting the sequence's behavior.

Exponents

Exponents are a powerful mathematical tool that indicate repeated multiplication of a number by itself. Consider \((-2)^{n+1}\) and \((-3)^n\) from our sequence formula. These are both instances of exponential expressions.

Here’s a quick rundown of exponents:

• Positive versus Negative Exponents: Positive exponents refer to multiplying the base by itself. Negative exponents mean repeated division or applying the inverse.


• Effects of Base Sign: When the base is negative, as in our formula, the exponent dictates the sign of the result. Odd exponents yield negative results for negative bases, while even exponents result in positive outcomes.


• Calculation Order: To handle exponents correctly in a calculation, you must first resolve exponents before any other operations such as addition or subtraction.

Understanding exponents enables precise computations in mathematical problems and helps in recognizing patterns within the sequence.

Arithmetic Sequences

An arithmetic sequence is one where each term is derived by adding a constant to the previous term. However, it's crucial to note that the sequence in our exercise \(a_n = (-2)^{n+1} + (-3)^n\) is not directly an arithmetic sequence, but understanding the concept of arithmetic sequences helps in recognizing when sequences deviate from this pattern.

Key characteristics of arithmetic sequences include:

• Common Difference: The difference between consecutive terms remains the same throughout the sequence.


• Simplicity of Pattern: The predictable addition or subtraction makes arithmetic sequences straightforward to predict and calculate.

In contrast, the given sequence involves exponential terms, which can cause greater variations between each subsequent term. Recognizing these differences helps in better understanding and analyzing complex sequences.