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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

User Zaptree
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1 Answer

7 votes

Answer:

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

User Cupid Chan
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