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

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