Question

for problems 1 - 3, create the missing equation for the sequence based on the table, the description, or the given equation. then decide which equation will help to find the missing term of the sequence most efficiently. find the term and explain your choice.
1.

term #	1	2	3	4
value	2	8	14

explicit equation:
find the 4th term:
story:

1. explicit equation: (f(n)=2n + 10)

term #	1	2		50
value	12	14

find the value of the 50th term:
story:

1. the value of the 6th term is 64.

the sequence is being doubled at each step.
explicit equation: (f(n)=1(2)^n)
find the value of the 7th term:
story:
evaluate the function to determine the output.
recall: (f(2)) means to replace the value of (n) with 2 and perform the operations to find (f(n)). the point ((n,f(n))) is the same location on the graph as ((x,y)), where (n) describes the location along the (x - axis) and (f(n)) is the height of the graph.
given that (f(n)=2(3^n)) and (g(n)=5(2^n)), evaluate the following functions with the indicated values.

1. (f(5)=)
2. (g(0)=)
3. (f(2)=)
4. (g(3)=)

Explanation:

Step1: Analyze the first sequence

We first find the common - difference of the sequence with terms 2, 8, 14. The common difference $d=8 - 2=6$. The first - term $a_1 = 2$. The explicit formula for an arithmetic sequence is $a_n=a_1+(n - 1)d$. Substituting $a_1 = 2$ and $d = 6$ into the formula, we get $a_n=2+(n - 1)\times6=2+6n-6=6n - 4$. To find the 4th term, we substitute $n = 4$ into the formula: $a_4=6\times4-4=24 - 4=20$.

Step2: Analyze the second sequence

Given the explicit equation $f(n)=2n + 10$. To find the 50th term, we substitute $n = 50$ into the formula. $f(50)=2\times50+10=100 + 10=110$.

Step3: Analyze the third sequence

Given the explicit equation $f(n)=1\times(2)^n$. To find the 7th term, we substitute $n = 7$ into the formula. $f(7)=1\times2^7=128$.

Step4: Evaluate $f(n)=2\times(3^n)$ for $n = 5$

Substitute $n = 5$ into $f(n)=2\times(3^n)$. $f(5)=2\times3^5=2\times243 = 486$.

Step5: Evaluate $g(n)=5\times(2^n)$ for $n = 0$

Substitute $n = 0$ into $g(n)=5\times(2^n)$. $g(0)=5\times2^0=5\times1 = 5$.

Step6: Evaluate $f(n)=2\times(3^n)$ for $n = 2$

Substitute $n = 2$ into $f(n)=2\times(3^n)$. $f(2)=2\times3^2=2\times9 = 18$.

Step7: Evaluate $g(n)=5\times(2^n)$ for $n = 3$

Substitute $n = 3$ into $g(n)=5\times(2^n)$. $g(3)=5\times2^3=5\times8 = 40$.

Answer:

1. Explicit equation: $a_n=6n - 4$, 4th term: 20
2. 50th - term value: 110
3. 7th - term value: 128
4. $f(5)=486$
5. $g(0)=5$
6. $f(2)=18$
7. $g(3)=40$