Carlos and Ana were asked to find an explicit formula for the sequence $-10,-4,2,8, \ldots$ , where the first term should be $f(1)$ . $\newline$ Carlos said the formula is $f(n)=-10+6 n$ . $\newline$ Ana said the formula is $f(n)=-10+6(n-1)$ . $\newline$ Which one of them is right? $\newline$ Choose $1$ answer: $\newline$ (A) Only Carlos $\newline$ (B) Only Ana $\newline$ (C) Both Carlos and Ana $\newline$ (D) Neither Carlos nor Ana

Full solution

Q. Carlos and Ana were asked to find an explicit formula for the sequence

-10,-4,2,8, \ldots

, where the first term should be

f(1)

\newline

Carlos said the formula is

f(n)=-10+6 n

\newline

Ana said the formula is

f(n)=-10+6(n-1)

\newline

Which one of them is right?

\newline

Choose

1

answer:

\newline

(A) Only Carlos

\newline

(B) Only Ana

\newline

\newline

(D) Neither Carlos nor Ana

Identify Sequence Type: Identify the type of sequence and the common difference. $\newline$ The sequence given is $-10, -4, 2, 8, \ldots$ which increases by $6$ each time. This indicates that it is an arithmetic sequence with a common difference $d$ of $6$ .
Write Explicit Formula: Write the general form of the explicit formula for an arithmetic sequence. $\newline$ The general form of the explicit formula for an arithmetic sequence is $f(n) = a_1 + (n-1)d$ , where $a_1$ is the first term and $d$ is the common difference.
Determine First Term: Determine the first term $a_1$ of the sequence. $\newline$ The first term given in the sequence is $-10$ , so $a_1 = -10$ .
Substitute Values: Substitute the values of $a_1$ and $d$ into the general formula. $\newline$ Substituting $a_1 = -10$ and $d = 6$ into the formula gives us $f(n) = -10 + (n-1)\times6$ .
Check Carlos's Formula: Check Carlos's formula. $\newline$ Carlos's formula is $f(n) = -10 + 6n$ . Let's check if this formula gives us the correct terms of the sequence. For $n=1$ , $f(1) = -10 + 6(1) = -4$ , which is not the first term of the sequence. Therefore, Carlos's formula is incorrect.
Check Ana's Formula: Check Ana's formula. $\newline$ Ana's formula is $f(n) = -10 + 6(n-1)$ . Let's check if this formula gives us the correct terms of the sequence. For $n=1$ , $f(1) = -10 + 6(1-1) = -10$ , which is the first term of the sequence. Therefore, Ana's formula is correct.