Algebra - Fibonacci, triangular, and other sequences

Find the $15^{th}$ triangular number ($T_{15}$).

$T_{15} = \frac{15(15 + 1)}{2}$ $T_{15} = \frac{15 \times 16}{2}$ $T_{15} = 15 \times 8 = 120$

To find any triangular number, use the formula $T_n = \frac{n(n+1)}{2}$ where $n$ is the position in the sequence.

A Fibonacci-type sequence begins with $u_1 = 3$ and $u_2 = 8$. Find the value of $u_5$.

$u_3 = 3 + 8 = 11$ $u_4 = 8 + 11 = 19$ $u_5 = 11 + 19 = 30$

In a Fibonacci sequence, each term is the sum of the two terms directly before it. We calculate $u_3$, then $u_4$, and finally $u_5$ using this recursive logic.

Determine the $n^{th}$ term formula for the sequence $7, 11, 15, 19, \dots$.

First term $u_1 = 7$. Common difference $d = 11 - 7 = 4$. $u_n = u_1 + (n - 1)d$ $u_n = 7 + (n - 1)4$ $u_n = 7 + 4n - 4$ $u_n = 4n + 3$

This is an arithmetic sequence because the difference between terms is constant ($d = 4$). We substitute the first term and the difference into the general arithmetic formula and simplify.