Algebra - Linear sequences

Find the $n^{th}$ term for the sequence: $5, 8, 11, 14, ...$

$3n + 2$

First, find the common difference: $8 - 5 = 3$, so $d = 3$. This gives us $3n$. Next, find the 'zero term' by subtracting the difference from the first term: $5 - 3 = 2$. Therefore, the $n^{th}$ term is $3n + 2$.

A sequence has the $n^{th}$ term rule $7n - 4$. Calculate the $50^{th}$ term.

346

To find the $50^{th}$ term, substitute $n = 50$ into the formula: $7(50) - 4 = 350 - 4 = 346$.

Determine if 100 is a term in the sequence $4n + 3$.

No

Set the formula equal to 100: $4n + 3 = 100$. Solve for $n$: $4n = 97$, so $n = 24.25$. Since $n$ must be a whole number (a position in the sequence), 100 is not a term in this sequence.

Find the $n^{th}$ term for the decreasing sequence: $20, 15, 10, 5, ...$

$-5n + 25$

The common difference is $15 - 20 = -5$. The zero term is $20 - (-5) = 25$. Thus, the rule is $-5n + 25$.