Predicting What Comes Next: Exploring Sequences - Explicit Rule for a Sequence

Write the first three terms of the sequence whose explicit rule is given by $a_n = 3n + 5$.

For $n = 1$: $a_1 = 3(1) + 5 = 8$ For $n = 2$: $a_2 = 3(2) + 5 = 11$ For $n = 3$: $a_3 = 3(3) + 5 = 14$

To find the first three terms, substitute the values $n=1$, $n=2$, and $n=3$ into the explicit formula.

Find the $15^{th}$ term of the sequence defined by the rule $a_n = \frac{n}{n + 1}$.

Substitute $n = 15$ into the formula: $a_{15} = \frac{15}{15 + 1} = \frac{15}{16}$

The explicit rule allows direct calculation of any term. By replacing $n$ with $15$, we find the value at that specific position.

Determine the explicit rule for the sequence: $4, 8, 12, 16, \dots$.

Observe the terms: $a_1 = 4 = 4 \times 1$ $a_2 = 8 = 4 \times 2$ $a_3 = 12 = 4 \times 3$ $a_4 = 16 = 4 \times 4$ Therefore, the rule is $a_n = 4n$.

By comparing the value of each term to its position $n$, we see that each term is $4$ times its position. This gives the rule $a_n = 4n$.

Find the difference between the $10^{th}$ term and the $8^{th}$ term of the sequence $a_n = n^2 + 2$.

First, find $a_{10}$: $a_{10} = 10^2 + 2 = 100 + 2 = 102$ Next, find $a_8$: $a_8 = 8^2 + 2 = 64 + 2 = 66$ Subtract the values: $\begin{array}{r} 102 \\ - 66 \\ \hline 36 \end{array}$

We calculate both specific terms using the explicit rule and then find their difference using vertical subtraction.