Answer:
The first term is 3 and common difference is 8.
And
The formula for sequence is: [tex]a_n = -5+8n[/tex]
Function is: [tex]f(n) = f(n-1)+8[/tex]
Step-by-step explanation:
Given that
[tex]a_4 = 27\\a_8 = 59[/tex]
The general formula for arithmetic sequence is:
[tex]a_n = a+(n-1)d[/tex]
Here
a is the first term
n is the term number
and d is the common difference
for 4th term
[tex]a_4 = a+(4-1)d\\27 = a+3d\ \ \ Eqn\ 1[/tex]
For 8th term
[tex]59 = a+ (8-1)d\\59= a+7d\ \ \ \ \ Eqn\ 2[/tex]
subtracting equation 1 from equation 2
[tex]59-27 = a+7d-(a+3d)\\32 = a+7d-a-3d\\32 = 4d\\d =\frac{32}{4}\\d = 8[/tex]
Putting d = 8 in equation 1
[tex]a+3d = 27\\a+3(8) = 27\\a+24=27\\a = 27-24\\a = 3[/tex]
Now for the function
[tex]a_n = a+(n-1)d\\a_n = 3+(n-1)(8)\\a_n = 3 + 8n-8\\a_n = -5+8n[/tex]
The sequence can also be expressed as a function as:
[tex]f(n) = f(n-1) + 8[/tex]
Hence,
The first term is 3 and common difference is 8.
And
The formula for sequence is: [tex]a_n = -5+8n[/tex]
Function is: [tex]f(n) = f(n-1)+8[/tex]
