Arithmetic Sequence: Is 29 the nth Term When a₁=1.5 and d=3?

Given a series whose first element is 1.5.

Each element of the series is greater by 3 of its predecessor.

Is the number 29 an element in the series?

If so, please indicate your place in the series.

To determine whether the number 29 is an element of the series, we start by recognizing that the problem involves an arithmetic sequence. In such a sequence, each term is generated by adding a constant difference to the previous term. Here, the first term $a_1$ is $1.5$, and the common difference $d$ is $3$.

The formula for the nth term of an arithmetic sequence is given by:

$a_n = a_1 + (n-1) \cdot d$

We need to check if 29 is one of the terms of this series, so we set $a_n = 29$ and solve for $n$:

$29 = 1.5 + (n-1) \cdot 3$

Subtract 1.5 from both sides:

$29 - 1.5 = (n-1) \cdot 3$

$27.5 = (n-1) \cdot 3$

Divide both sides by 3 to solve for $n - 1$:

$n-1 = \frac{27.5}{3}$

$n-1 = 9.1666\ldots$

Add 1 to find $n$:

$n = 10.1666\ldots$

Since $n$ is not an integer, the number 29 does not appear as an element in this sequence. Arithmetic sequences only have integer positions for their terms, so $n$ must be a whole number for 29 to be a term. As a result, the correct answer is:

No