Respuesta :
Answer:
[tex]a_n=22-4n[/tex]
Step-by-step explanation:
We are asked to find the formula of arithmetic sequence, whose 3rd term is 10 and 51st term is [tex]-182[/tex].
We know that an arithmetic sequence is in format [tex]a_n=a_1+(n-1)d[/tex], where,
[tex]a_n[/tex] = nth term of sequence,
[tex]a_1[/tex] = 1st term of sequence,
n = Number terms in the sequence.
d = Common difference.
Upon substituting our given values, we will get two equation as:
[tex]a_3=a_1+(3-1)d...(1)[/tex]
[tex]10=a_1+2d...(1)[/tex]
[tex]a_51=a_1+(51-1)d...(2)[/tex]
[tex]-182=a_1+50d...(2)[/tex]
Subtracting equation (2) from equation (1), we will get:
[tex]10-(-182)=a_1-a_1+2d-50d[/tex]
[tex]10+182=-48d[/tex]
[tex]192=-48d[/tex]
Switch sides:
[tex]-48d=192[/tex]
[tex]\frac{-48d}{-48}=\frac{192}{-48}[/tex]
[tex]d=-4[/tex]
Upon substituting [tex]d=-4[/tex]in equation (1), we will get:
[tex]10=a_1+2(-4)[/tex]
[tex]10=a_1-8[/tex]
[tex]10+8=a_1-8+8[/tex]
[tex]18=a_1[/tex]
Upon substituting [tex]a_1=18[/tex] and [tex]d=-4[/tex] in arithmetic sequence formula, we will get:
[tex]a_n=18+(n-1)-4[/tex]
[tex]a_n=18-4n+4[/tex]
[tex]a_n=22-4n[/tex]
Therefore, our required formula would be [tex]a_n=22-4n[/tex].
