Imran and Aubrey were asked to find an explicit formula for the sequence $14,5,-4,-13, \ldots$ , where the first term should be $g(1)$ . $\newline$ Imran said the formula is $g(n)=14-9(n-1)$ . $\newline$ Aubrey said the formula is $g(n)=14-9 n$ . $\newline$ Which one of them is right? $\newline$ Choose $1$ answer: $\newline$ (A) Only Imran $\newline$ (B) Only Aubrey $\newline$ (C) Both Imran and Aubrey $\newline$ (D) Neither Imran nor Aubrey

Full solution

Q. Imran and Aubrey were asked to find an explicit formula for the sequence

14,5,-4,-13, \ldots

, where the first term should be

g(1)

\newline

Imran said the formula is

g(n)=14-9(n-1)

\newline

Aubrey said the formula is

g(n)=14-9 n

\newline

Which one of them is right?

\newline

Choose

1

answer:

\newline

(A) Only Imran

\newline

(B) Only Aubrey

\newline

\newline

(D) Neither Imran nor Aubrey

Identify type of sequence: Identify the type of sequence. The sequence $14, 5, -4, -13, ext{...}$ has a constant difference between terms, which makes it an arithmetic sequence.
Determine common difference: Determine the common difference $d$ of the sequence. The difference between the first term $14$ and the second term $5$ is $5 - 14 = -9$ . This is the common difference.
Use arithmetic sequence formula: Use the arithmetic sequence formula to find the nth term: $g(n) = g(1) + (n-1)d$ . Here, $g(1)$ is the first term, which is $14$ , and $d$ is the common difference, which we found to be $-9$ .
Substitute values into formula: Substitute the known values into the formula to get Imran's proposed formula: $g(n) = 14 + (n-1)(-9)$ . Simplify the formula to get $g(n) = 14 - 9(n-1)$ .
Check Imran's formula: Check Imran's formula by plugging in $n=2$ to see if it gives the second term of the sequence: $g(2) = 14 - 9(2-1) = 14 - 9 = 5$ . This matches the second term of the sequence, so Imran's formula seems correct so far.
Check Aubrey's formula: Now, check Aubrey's formula by plugging in $n=1$ to see if it gives the first term of the sequence: $g(1) = 14 - 9(1) = 14 - 9 = 5$ . This does not match the first term of the sequence, which should be $14$ . Therefore, Aubrey's formula is incorrect.
Correct answer: Since Imran's formula gives the correct terms for the sequence and Aubrey's does not, the correct answer is that only Imran is right.