Question

Find the first term and the common difference of the arithmetic sequence described. Give a recursive formula for the sequence. Find a formula for the nth term. 5 th term is 7​; 19 th term is 63

Answer 1

First Term:
`a_(1) = -9`

Common Difference : d= 4

Recursive formula:
`a_(n) = -9 + 4(n-1)`

Explanation:

Here we are going to use the arithmetic progression formula:

`a_(n) = a_(1) + (n-1) d`

`a_(n) : nth - term`

d :common difference

Since we are given the 5th term and 19th term, we can write them as :

`a_(5) = a_(1) + (5-1) d`

`a_(5) = a_(1) + (4) d`

as
`a_(5)=7`

so,

`7 = a_(1) + (4) d` -------------------(Equation 1)

Moreover, using same formula for 19th term

`a_(19) = a_(1) + (19-1) d`

`a_(19) = a_(1) + (18) d`

As
`a_(19) = 63`

So,

`63=a_(1) + (18) d` --------------------(Equation 2)

From Equation 1, we have:

`a_(1) = 7-4d`

put the value in equation 2:

`63= 7-4d+18d`

`63=7+14d`

`63-7 = 14d\\14d = 56\\d= (56)/(14) \\d=4`

Which is the common difference

Now put the value of d in equation 1:

`7= a_(1) +4 (4)\\7=a_(1) +16\\7-16 = a_(1) \\`

`a_(1) = -9`

Which is the first term

Putting the firm term and common difference in the initial arithmetic progression formula:

`a_(n) = a_(1) + (n-1) d`

`a_(n)= -9 + (n-1)d \\`

`a_(n)= -9 + 4(n-1)`

Which is the recursive formula of the sequence