Question

find $a_{7}.$

1. $\\{2, 3, 8, 19, 46, ...\\}$

$a_1=2$
$a_n=$

Explanation:

Step1: Identify recurrence relation

Observe the sequence:
$3 = 2\times2 - 1$
$8 = 3\times2 + 2$
$19 = 8\times2 + 3$
$46 = 19\times2 + 8$
The pattern is $a_n = 2a_{n-1} + a_{n-2}$ for $n\geq3$, with $a_1=2$, $a_2=3$.

Step2: Calculate $a_6$

Use the recurrence relation:
$a_6 = 2a_5 + a_4 = 2\times46 + 19 = 92 + 19 = 111$

Step3: Calculate $a_7$

Use the recurrence relation:
$a_7 = 2a_6 + a_5 = 2\times111 + 46 = 222 + 46 = 268$

Answer:

$a_7=268$