Question

question
the first three terms of a sequence are given. write your answer as a decimal or whole number.
round to the nearest thousandth (if necessary).
18, 30, 50, ...
find the 8th term.

Explanation:

Step1: Find second differences

First differences: $30-18=12$, $50-30=20$
Second difference: $20-12=8$

Step2: Identify sequence type

Since second differences are constant ($8$), this is a quadratic sequence with general form $a_n = an^2 + bn + c$.

Step3: Set up system of equations

For $n=1$: $a(1)^2 + b(1) + c = 18$ → $a + b + c = 18$
For $n=2$: $a(2)^2 + b(2) + c = 30$ → $4a + 2b + c = 30$
For $n=3$: $a(3)^2 + b(3) + c = 50$ → $9a + 3b + c = 50$

Step4: Solve for $a$

Subtract first equation from second:
$(4a + 2b + c) - (a + b + c) = 30 - 18$ → $3a + b = 12$
Subtract second equation from third:
$(9a + 3b + c) - (4a + 2b + c) = 50 - 30$ → $5a + b = 20$
Subtract these two results:
$(5a + b) - (3a + b) = 20 - 12$ → $2a = 8$ → $a = 4$

Step5: Solve for $b$

Substitute $a=4$ into $3a + b = 12$:
$3(4) + b = 12$ → $12 + b = 12$ → $b = 0$

Step6: Solve for $c$

Substitute $a=4, b=0$ into $a + b + c = 18$:
$4 + 0 + c = 18$ → $c = 14$

Step7: Define nth term formula

$a_n = 4n^2 + 14$

Step8: Calculate 8th term

Substitute $n=8$:
$a_8 = 4(8)^2 + 14 = 4(64) + 14$

Answer:

270