Recursive formula: [tex]a_n = a_{n-1} -6[/tex]
Explicit Formula: [tex]a_n = 34-6n[/tex]
35th term: -176
Sum of first 35 terms: -2590
Step-by-step explanation:
Given sequence is:
28, 22, 16, 10
First of all, we have to determine if this is an arithmetic sequence or geometric sequence. for this purpose we will find the common difference.
Here
[tex]a_1 = 28\\a_2 = 22\\a_3 = 16[/tex]
[tex]d = a_2-a_1 = 22-28 =-6\\d = a_3-a_2 = 16-22 = -6[/tex]
As the common difference is same the sequence is an arithmetic sequence
Recursive Formula:
Recursive formula is the formula that uses the previous term and common difference to find the next term
As the common difference is -6
The recursive formula will be:
[tex]a_n = a_{n-1} +(-6)\\a_n = a_{n-1} -6[/tex]
Explicit formula:
The explicit formula for an arithmetic sequence is:
[tex]a_n = a_1+(n-1)d[/tex]
Putting the value of d and a_1
[tex]a_n = 28 + (n-1)(-6)\\= 28 -6n+6\\= 34-6n[/tex]
35th term of the sequence:
To find the 35th term, putting n = 35
[tex]a_{35} = 34-6(35)\\a_{35} = 34 - 210\\a_{35} = -176[/tex]
Sum:
[tex]a_1 = 28\\a_{35} = -176[/tex]
The formula for sum of finite arithmetic sequence is:
[tex]S_n = \frac{n}{2} (a_1+a_n)[/tex]
As we have to find sum of 35 terms, the formula will be:
[tex]S_{35} = \frac{35}{2}(28-176)\\S_{35} = 17.5 (-148)\\=-2590[/tex]
Hence,
Recursive formula: [tex]a_n = a_{n-1} -6[/tex]
Explicit Formula: [tex]a_n = 34-6n[/tex]
35th term: -176
Sum of first 35 terms: -2590
Keywords: Arithmetic sequence, common difference
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