In a certain arithmetic sequence, the 10th term is equal to 10 and the 12th term is equal to 100. What must the 11th term be?

Question

I really need help with this!

Answer 1

There is a shortcut you can use, and I will toss it your way! Note that you jumped from 10 to 100 in 2 terms. All you need is to find the slope of this line, which is:

100 - 10        90

----------  = ------  = 45

      2             2

 

That represents the change in each term, which is represented by "d" in the formula. If the 10th term is 10, the 11th will be 10+45 = 55 (and 55+45 = 100, exactly as described).

Answer 2

In an arithmetic sequence numbers increase or decrease at a constant rate.  In this problem we are going from 10 to 100 in two steps, so the 11th term would be halfway between the two.

 

100 - 10 = 90

90/2 = 45

10 + 45 = 55

 

The 11th term must be 55.

Answer 3

a₁₀ = 10

a₁₂ = 100

 

a₁₂ = a₁₀ + 2d

100 = 10 + 2d

90 = 2d

45 = d

a₁₁ = 10 + 45

 

Or another way

 

a₁₁ = (a₁₀ + a₁₂) / 2

a₁₁ = 110 / 2

a₁₁ = 55