Question

Vera and Fenyang were asked to find an explicit formula for the sequence
\[26,10,-6,-22,\unicode{0x2026}\], where the first term should be
\[g(1)\].
Vera said the formula is
\[g(n)=26-16(n-1)\].
Fenyang said the formula is
\[g(n)=42-16n\].
Which one of them is right?
Choose 1 answer:
Choose 1 answer:
(Choice A) Only Vera
A
Only Vera
(Choice B) Only Fenyang
B
Only Fenyang
(Choice C) Both Vera and Fenyang
C
Both Vera and Fenyang
(Choice D) Neither Vera nor Fenyang
D
Neither Vera nor Fenyang

Answer 1

To determine which of Vera's or Fenyang's formula is correct for the sequence \(26, 10, -6, -22, \ldots\), we first need to recognize the pattern of the sequence.

Calculating the differences between each term:
- From \(26\) to \(10\): \(10 - 26 = -16\)
- From \(10\) to \(-6\): \(-6 - 10 = -16\)
- From \(-6\) to \(-22\): \(-22 - (-6) = -16\)

Thus, the differences between consecutive terms are consistent at \(-16\). This indicates that the sequence is arithmetic with a first term \(g(1) = 26\) and a common difference of \(-16\).

The general formula for an arithmetic sequence can be expressed as:
\[
g(n) = g(1) + (n-1)d
\]
where \(g(1)\) is the first term, \(d\) is the common difference, and \(n\) is the term number.

Substituting in the known values:
- \(g(1) = 26\)
- \(d = -16\)

Thus, we have:
\[
g(n) = 26 + (n - 1)(-16) = 26 - 16(n - 1)
\]

This matches Vera's formula:
\[
g(n) = 26 - 16(n - 1)
\]

Next, let’s analyze Fenyang’s formula:
\[
g(n) = 42 - 16n
\]
Calculating a few terms using Fenyang's formula:
- For \(n = 1\): \(g(1) = 42 - 16 \cdot 1 = 42 - 16 = 26\)
- For \(n = 2\): \(g(2) = 42 - 16 \cdot 2 = 42 - 32 = 10\)
- For \(n = 3\): \(g(3) = 42 - 16 \cdot 3 = 42 - 48 = -6\)
- For \(n = 4\): \(g(4) = 42 - 16 \cdot 4 = 42 - 64 = -22\)

Fenyang's formula also produces the correct terms for the sequence. Therefore, both Vera and Fenyang have valid formulas.

Given that both Vera and Fenyang provided correct formulas for the same sequence, the answer is:

**(Choice C) Both Vera and Fenyang**