Question

first term of a sequence is 3 what are the remaining terms to be found using formula a↓n = 5a↓(n – 1) – 2

Answer 1

Given the first term \( a_1 = 3 \) and the recurrence relation \( a_n = 5a_{n-1} - 2 \), we can find the remaining terms of the sequence.

Using the formula, we calculate each term step by step:

1. **First term**:
\[
a_1 = 3
\]

2. **Second term** \( a_2 \):
\[
a_2 = 5a_1 - 2 = 5 \cdot 3 - 2 = 15 - 2 = 13
\]

3. **Third term** \( a_3 \):
\[
a_3 = 5a_2 - 2 = 5 \cdot 13 - 2 = 65 - 2 = 63
\]

4. **Fourth term** \( a_4 \):
\[
a_4 = 5a_3 - 2 = 5 \cdot 63 - 2 = 315 - 2 = 313
\]

5. **Fifth term** \( a_5 \):
\[
a_5 = 5a_4 - 2 = 5 \cdot 313 - 2 = 1565 - 2 = 1563
\]

Now we can summarize the first five terms of the sequence:

- \( a_1 = 3 \)
- \( a_2 = 13 \)
- \( a_3 = 63 \)
- \( a_4 = 313 \)
- \( a_5 = 1563 \)

Thus, the terms of the sequence are:
\[
3, 13, 63, 313, 1563
\]