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

Answer:

First Term: $a_{1} = -9$

Common Difference : d= 4

Recursive formula:$a_{n} = -9 + 4(n-1)$

Step-by-step 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:

$63= 7-4d+18d$

$63=7+14d$

$63-7 = 14d\\14d = 56\\d= \frac{56}{14} \\d=4$

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