Question

The recursive equation of a sequence is given by a Subscript n Baseline equals a Subscript n minus 1 Baseline plus 7 for n is greater than or equal to 1.



If a Subscript 5 Baseline equals 43, what is a Subscript 1 Baseline?

Answer options with 5 options
A.
1

B.
6

C.
8

D.
10

E.
15

Answer 1

To solve for \( a_1 \) given the recursive equation \( a_n = a_{n-1} + 7 \) for \( n \geq 1 \) and the condition \( a_5 = 43 \), we can work backwards through the sequence.

1. **Starting from \( a_5 \)**:
\[
a_5 = a_4 + 7
\]
Since \( a_5 = 43 \):
\[
43 = a_4 + 7 \implies a_4 = 43 - 7 = 36
\]

2. **Now calculate \( a_4 \)**:
\[
a_4 = a_3 + 7
\]
From above, \( a_4 = 36 \):
\[
36 = a_3 + 7 \implies a_3 = 36 - 7 = 29
\]

3. **Calculate \( a_3 \)**:
\[
a_3 = a_2 + 7
\]
From above, \( a_3 = 29 \):
\[
29 = a_2 + 7 \implies a_2 = 29 - 7 = 22
\]

4. **Calculate \( a_2 \)**:
\[
a_2 = a_1 + 7
\]
From above, \( a_2 = 22 \):
\[
22 = a_1 + 7 \implies a_1 = 22 - 7 = 15
\]

Thus, we find that \( a_1 = 15 \).

The answer is \( \boxed{15} \).