Question

What is the 5th term of an arithmetic sequence if t2 = -5 and t6 = 7?

Answer 1

The fifth term is 4.

Explanation:

The model for an arithmetic sequence is a(n) = a(1) + d(n -1), where d is the common difference. Using the given information, we get:

a(2) = a(1) + d(2 - 1) = -5, or a(1) + d = -5

a(6) = a(1) + d(6 - 1) = 7, or a(1) + 5d) = 7

Here we have two equations in two unknowns: a(1) and d.

Subtract the first equation from the second:

a(1) + 5d = 7

-a(1) - d = 5

--------------------

4d = 12

Thus, d = 12/4 = 3. This is the common difference.

Use the equation a(1) + d = -5 (from above) to find the value of a(1):

a(1) + 3 = -5, or a(1) = -8

Then the equation for this arithmetic sequence is a(n) = -8 + 3(n - 1).

The 5th term of this sequence is thus

a(5) = -8 + 3(5 - 1), or a(5) = -8 + 3(4), or -8 + 12, or 4.

Answer 2

Explanation:

Let the first term of the arithmetic sequence be a and common difference be d

Then, t2 = -5

or, a+(n-1)d= -5

or, a+(2-1)d= -5

or,a+ d = -5 ---------------(i)

Again, t6= 7

or, a+(6-1)d= 7

or, a + 5d = 7------------(ii)

Now equation (ii) - (i),

a+5d - ( a+d)= 7-(-5)

a+5d-a-d = 7+5

4d= 12

d = 3

From equation (i),

a +d=-5

a+3= -5

a=-5-3 = -8

t5 = a+(5-1)d

= -8+4(3)

= -8+12

= 4