Question

Ivy and Andrey were asked to find an explicit formula for the sequence
\[-100,-50,0,50,\unicode{0x2026}\], where the first term should be
\[f(1)\].
Ivy said the formula is
\[f(n)=-100+50(n-1)\].
Andrey said the formula is
\[f(n)=-150+50n\].
Which one of them is right?
Choose 1 answer:
Choose 1 answer:
(Choice A) Only Ivy
A
Only Ivy
(Choice B) Only Andrey
B
Only Andrey
(Choice C) Both Ivy and Andrey
C
Both Ivy and Andrey
(Choice D) Neither Ivy nor Andrey
D
Neither Ivy nor Andrey

Answer 1

To determine which formula is correct for the sequence \(-100, -50, 0, 50, \ldots\), we can analyze both Ivy's and Andrey's formulas.

1. **Ivy's formula**:
\[
f(n) = -100 + 50(n - 1)
\]
Let's compute the first few terms using her formula:
- For \(n = 1\):
\[
f(1) = -100 + 50(1 - 1) = -100 + 0 = -100
\]
- For \(n = 2\):
\[
f(2) = -100 + 50(2 - 1) = -100 + 50 = -50
\]
- For \(n = 3\):
\[
f(3) = -100 + 50(3 - 1) = -100 + 100 = 0
\]
- For \(n = 4\):
\[
f(4) = -100 + 50(4 - 1) = -100 + 150 = 50
\]
This matches the sequence.

2. **Andrey's formula**:
\[
f(n) = -150 + 50n
\]
Let's compute the first few terms using his formula:
- For \(n = 1\):
\[
f(1) = -150 + 50(1) = -150 + 50 = -100
\]
- For \(n = 2\):
\[
f(2) = -150 + 50(2) = -150 + 100 = -50
\]
- For \(n = 3\):
\[
f(3) = -150 + 50(3) = -150 + 150 = 0
\]
- For \(n = 4\):
\[
f(4) = -150 + 50(4) = -150 + 200 = 50
\]
This also matches the sequence.

Both Ivy and Andrey provided correct formulas that produce the same sequence. Therefore, the correct answer is:

**(Choice C) Both Ivy and Andrey**