Algebra - General Rules and Formulas for Sequences

Find the $n^{th}$ term rule for the sequence: $7, 11, 15, 19, \dots$

$u_n = 4n + 3$

First, find the common difference $d$: $11 - 7 = 4$. This means the rule starts with $4n$. To find the constant $c$, look at the first term ($n=1$): $4(1) + c = 7$, which gives $c = 3$. Alternatively, subtract the difference from the first term: $7 - 4 = 3$.

For the sequence defined by $u_n = 5n - 2$, calculate the $25^{th}$ term.

$u_{25} = 123$

Substitute $n = 25$ into the formula: $u_{25} = 5(25) - 2$. Calculating this gives $125 - 2 = 123$.

Determine the first three terms of a geometric sequence where the first term $a = 3$ and the common ratio $r = 2$.

$3, 6, 12$

The first term $u_1 = 3$. The second term $u_2 = 3 \times 2 = 6$. The third term $u_3 = 6 \times 2 = 12$.

Is the number $50$ a term in the sequence $u_n = 3n + 5$?

$n = 15$

Set the formula equal to $50$: $3n + 5 = 50$. Subtract $5$ from both sides: $3n = 45$. Divide by $3$: $n = 15$. Since $15$ is a whole number, $50$ is the $15^{th}$ term of the sequence.