Math Problem Statement

Solve this

Solve the sequence \( a_n = \frac{16^{n+1}}{4^n + 8 \cdot 16^n} \).

Solution

The expression provided is:

an=16n+14n+816na_n = \frac{16^{n+1}}{4^n + 8 \cdot 16^n}

To simplify, let’s express 4n4^n and 16n16^n in terms of powers of 2.

  1. Rewrite powers in terms of base 2:
    • Since 16=2416 = 2^4, then 16n=(24)n=24n16^n = (2^4)^n = 2^{4n}.
    • Similarly, 16n+1=24(n+1)=24n+416^{n+1} = 2^{4(n+1)} = 2^{4n + 4}.
    • For 4n4^n, we know 4=224 = 2^2, so 4n=(22)n=22n4^n = (2^2)^n = 2^{2n}.

Now the expression becomes:

an=24n+422n+824na_n = \frac{2^{4n+4}}{2^{2n} + 8 \cdot 2^{4n}}

  1. Rewrite the denominator: Notice that 8=238 = 2^3, so we have:

    an=24n+422n+2324n=24n+422n+24n+3a_n = \frac{2^{4n+4}}{2^{2n} + 2^3 \cdot 2^{4n}} = \frac{2^{4n+4}}{2^{2n} + 2^{4n+3}}

  2. Factor out 22n2^{2n} from the denominator:

    an=24n+422n(1+22n+3)a_n = \frac{2^{4n+4}}{2^{2n}(1 + 2^{2n+3})}

  3. Simplify the fraction: Divide the numerator and the denominator by 22n2^{2n}:

    an=24n+42n1+22n+3=22n+41+22n+3a_n = \frac{2^{4n+4 - 2n}}{1 + 2^{2n+3}} = \frac{2^{2n+4}}{1 + 2^{2n+3}}

At this point, further simplification might depend on the value of nn, but this is the simplified form in terms of exponents.

Would you like further explanation on this solution, or have additional questions?

Related Questions:

  1. How do powers of different bases relate in terms of simplification?
  2. What techniques can we use to simplify expressions with multiple exponential terms?
  3. How does factoring help in simplifying algebraic expressions?
  4. How can we apply these techniques to solve limits involving exponential functions?
  5. What are the properties of exponents useful for this problem?

Tip:

When simplifying expressions with exponents, converting all terms to a common base (like powers of 2 in this case) often makes the process more manageable.

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Math Problem Analysis

Mathematical Concepts

Algebra
Exponents and Powers
Simplification

Formulas

Conversion of bases for exponents: \( 16 = 2^4 \) and \( 4 = 2^2 \)
Simplifying expressions by factoring and common terms

Theorems

Laws of Exponents
Factoring in Algebra

Suitable Grade Level

Grade 10-12