Question

the first two terms of an arithmetic sequence are a1 = 2 and a2 = 5. find a6, the sixth term of the sequence.
a) 27
b) 20
c) 14
d) 17
question 6 (5 points)
identify the common ratio of the sequence: 400, 200, 100, 50, ...
a) 2
b) 1/4
c) 1/2

Explanation:

Step1: Find the common - difference of the arithmetic sequence

The formula for the common - difference $d$ of an arithmetic sequence is $d=a_{n + 1}-a_{n}$. Given $a_{1}=2$ and $a_{2}=5$, then $d=a_{2}-a_{1}=5 - 2=3$.

Step2: Use the formula for the $n$th term of an arithmetic sequence

The formula for the $n$th term of an arithmetic sequence is $a_{n}=a_{1}+(n - 1)d$. We want to find $a_{6}$, where $a_{1}=2$, $n = 6$, and $d = 3$. Substitute these values into the formula: $a_{6}=a_{1}+(6 - 1)d=2+5\times3=2 + 15=17$.

Step3: Find the common ratio of the geometric sequence

The formula for the common ratio $r$ of a geometric sequence is $r=\frac{a_{n + 1}}{a_{n}}$. For the sequence $400,200,100,50,\cdots$, take $a_{1}=400$ and $a_{2}=200$. Then $r=\frac{a_{2}}{a_{1}}=\frac{200}{400}=\frac{1}{2}$.

Answer:

D. 17
C. $\frac{1}{2}$