The fifth term of an arithmetic sequence is 23 and the 12th term is 72. And they say first question 1.determine the first three terms of the sequences and the nth term. 2.what is the value of the 10th term.

3. Which term has a value of 268.

3 answers

This is a routine arithmetic sequence question.
You MUST know the formulas for general term n, and the sum of n terms

if a is the first term, and d is the common difference,

term(n) = a + d(n-1)
sum(n) = (n/2)(2a + d(n-1)) or sum(n) = (n/2)(first + last)

here we have:
a+4d = 23
a+11d = 72
subtract them:
7d = 49
d = 7
sub back into 1st,
a + 28 = 23
a = -5

now:
first 3 terms are : -5, 2, 9
term(n) = a + d(n-1)
= -5 + 7(n-1)
= 7n - 12

we could now find any term using that general formula
term(10) = 7(10) - 12 = 58
notice we could have found this using our original formul
term(10) = a + 9d
= -5 + 63 = 58

7n - 12 = 268
7n = 280
n = 40
It is the 40th term

or

a+d(n-1) = 268
-5 + 7(n-1) = 268
-5 + 7n - 7 = 268
wow, we are getting the same steps.

-5,2,9

Thanks Same story