Question

for each sequence, determine whether it is arithmetic or geometric. also, write the formula for the n th term, a_n, of that sequence. (a) classify the sequence and write its n th term. 5, 12, 19, … the sequence is select arithmetic geometric a_n = \square (b) classify the sequence and write its n th term. 6, 12, 24, … the sequence is select a_n = \square

Explanation:

Step1: Classify sequence (a)

Check differences: $12-5=7$, $19-12=7$. Constant difference means arithmetic.

Step2: Find $n^\text{th}$ term for (a)

Use arithmetic formula $a_n = a_1 + (n-1)d$. Here $a_1=5$, $d=7$.
$a_n = 5 + (n-1) \times 7 = 7n - 2$

Step3: Classify sequence (b)

Check ratios: $\frac{12}{6}=2$, $\frac{24}{12}=2$. Constant ratio means geometric.

Step4: Find $n^\text{th}$ term for (b)

Use geometric formula $a_n = a_1 r^{n-1}$. Here $a_1=6$, $r=2$.
$a_n = 6 \times 2^{n-1} = 3 \times 2^n$

Answer:

(a) The sequence is Arithmetic
$a_n = 7n - 2$
(b) The sequence is Geometric
$a_n = 6 \times 2^{n-1}$ (or $3 \times 2^n$)