Question

Find the arithmetic sequence of $2, 5, 10, 17, 26$.

A) $aₙ = n² + 1$
B) $aₙ = 2n + 1$
C) $aₙ = n² + 2$
D) $aₙ = 2n² - 1$

Answer 1

The correct formula for the given sequence $2, 5, 10, 17, 26$ is A) $a_n = n^2 + 1$. This sequence is not strictly arithmetic, but option A correctly represents the general term for the given sequence.

Step-by-step explanation:

To find the arithmetic sequence for the given series $2, 5, 10, 17, 26$, we need to find the formula that describes the general term $a_n$. By examining the given sequence, we can recognize that the differences between terms are not constant, which means the sequence is not strictly arithmetic; it could, however, be described by a different formula.

Let's check each formula option given:

1. A) $a_n = n^2 + 1$
2. B) $a_n = 2n + 1$
3. C) $a_n = n^2 + 2$
4. D) $a_n = 2n^2 - 1$

If we plug in $n=1$ into option A ($n^2+1$), we get $1+1=2$, which matches the first number in the sequence. Similarly, for $n=2$, option A gives us $4+1=5$, and so on for the rest of the terms. This indicates that option A might be the correct formula.

To confirm, we should check the remaining terms in the sequence. If we continue this process, we will see that indeed every term in the sequence matches the output of option A for consecutive values of $n$. Therefore, the formula $a_n = n^2 + 1$ gives the correct terms for the sequence, indicating it represents the general term.