In Exercises 1–12, write the first four terms of each sequence whose general term is given. a_n=(−1)ⁿ⁺¹/(2ⁿ−1)

Verified step by step guidance

1. Identify the general term of the sequence: $a{n}_{}=\frac{{(-1)}^{\mathrm{n+1}}}{}$. Note that the expression is a fraction with numerator $(-1)^\{n+1\}$ and denominator $2^n\; -\; 1$.
2. Calculate the first term by substituting $n=1$ into the general term: $a\_1\; =\; \backslash frac\{(-1)^\{1+1\}\}\{2^1\; -\; 1\}$. Simplify the exponent and denominator accordingly.
3. Calculate the second term by substituting $n=2$ into the general term: $a\_2\; =\; \backslash frac\{(-1)^\{2+1\}\}\{2^2\; -\; 1\}$. Simplify the exponent and denominator accordingly.
4. Calculate the third term by substituting $n=3$ into the general term: $a\_3\; =\; \backslash frac\{(-1)^\{3+1\}\}\{2^3\; -\; 1\}$. Simplify the exponent and denominator accordingly.
5. Calculate the fourth term by substituting $n=4$ into the general term: $a\_4\; =\; \backslash frac\{(-1)^\{4+1\}\}\{2^4\; -\; 1\}$. Simplify the exponent and denominator accordingly.

Key Concepts

Sequences and General Terms

A sequence is an ordered list of numbers defined by a general term formula, an, which gives the nth term. Understanding how to substitute values of n into the formula allows you to find specific terms in the sequence.

Exponents and Powers

Exponents indicate repeated multiplication, such as 2^n meaning 2 multiplied by itself n times. Correctly evaluating powers is essential when calculating terms involving expressions like 2^n in the denominator.

Order of Operations and Simplification

Applying the correct order of operations (PEMDAS) ensures accurate evaluation of expressions, especially when dealing with powers, addition, and division in the formula. Simplifying each term carefully helps in writing the sequence terms correctly.