Certainly! Let's solve this problem step by step.
We are given a recursive sequence defined by \( a(n+1) = 5a(n) \) with the initial term \( a(1) = 2 \). Our task is to find the first five terms of this sequence.
### Step-by-Step Solution:
Step 1: Identify the Initial Term
The initial term of the sequence is given by:
[tex]\[ a(1) = 2 \][/tex]
Step 2: Apply the Recursive Relation to Find Subsequent Terms
The recursive relation tells us how to find the next term in the sequence from the current term. Specifically, each term is 5 times the previous term.
[tex]\[ a(n+1) = 5a(n) \][/tex]
Step 3: Calculate the Next Term
We'll start with the initial term and apply the recursive relation in succession.
- For \( n = 1 \):
[tex]\[ a(2) = 5a(1) = 5 \cdot 2 = 10 \][/tex]
- For \( n = 2 \):
[tex]\[ a(3) = 5a(2) = 5 \cdot 10 = 50 \][/tex]
- For \( n = 3 \):
[tex]\[ a(4) = 5a(3) = 5 \cdot 50 = 250 \][/tex]
- For \( n = 4 \):
[tex]\[ a(5) = 5a(4) = 5 \cdot 250 = 1250 \][/tex]
### Summary of the First Five Terms
We can now summarize the first five terms of the sequence:
[tex]\[ a(1) = 2 \][/tex]
[tex]\[ a(2) = 10 \][/tex]
[tex]\[ a(3) = 50 \][/tex]
[tex]\[ a(4) = 250 \][/tex]
[tex]\[ a(5) = 1250 \][/tex]
Thus, the sequence starts with the terms: \( [2, 10, 50, 250, 1250] \).
This concludes our detailed, step-by-step solution!
