Math Problem Statement

If a, start subscript, 1, end subscript, equals, 3a1​=3 and a, start subscript, n, end subscript, equals, 2, a, start subscript, n, minus, 1, end subscriptan​=2an−1​ then find the value of a, start subscript, 6, end subscripta6​

Solution

Given the recursive sequence defined by:

a1=3a_1 = 3 an=2an1for n>1a_n = 2a_{n-1} \quad \text{for } n > 1

We need to find the value of a6a_6.

Let's calculate each term up to a6a_6 step by step:

  1. Finding a2a_2:

a2=2a1=2×3=6a_2 = 2a_1 = 2 \times 3 = 6

  1. Finding a3a_3:

a3=2a2=2×6=12a_3 = 2a_2 = 2 \times 6 = 12

  1. Finding a4a_4:

a4=2a3=2×12=24a_4 = 2a_3 = 2 \times 12 = 24

  1. Finding a5a_5:

a5=2a4=2×24=48a_5 = 2a_4 = 2 \times 24 = 48

  1. Finding a6a_6:

a6=2a5=2×48=96a_6 = 2a_5 = 2 \times 48 = 96

So, the value of a6a_6 is:

96\boxed{96}

Would you like more details or have any questions about this solution?

Here are 5 related questions to explore:

  1. What is the value of a7a_7 in this sequence?
  2. Can you derive a general formula for ana_n based on the given recursive relation?
  3. How does the sequence change if the initial value a1a_1 is different?
  4. What type of sequence is this, and what are its general properties?
  5. How would the sequence be affected if the multiplication factor in the recursive relation was 3 instead of 2?

Tip: For geometric sequences, the nth term can often be expressed directly using the initial term and the common ratio raised to a power.