"Bailey Broker leases a property to Susie Smiley for 5 years. Bailey will receive a 5% commission. The rent for the property is $300 per month for the first year, with a $50-per-month increase each succeeding year. Knowing this, what will Bailey's commission be?

  $1200

Step-by-step explanation:

On an annual basis, Susie will be paying ($300×12) = $3600 the first year, and ($50×12) = $600 more each year after. The average rent over the 5-year period will be the rent in year 3, or $3600 +2×600 = $4800.

Susie's total rent for the period is ...

  $4800×5 = $24,000

and Bailey's total commission will be ...

  0.05 × $24,000 = $1200

The sequence of annual rents is an arithmetic sequence with first term 3600 and common difference 600. The total of n terms of an arithmetic sequence will be ...

  sum = (2·a1 +d(n -1))×(n/2)

  = (2·3600 +600(5-1))(5/2) = 24000 . . . for n=5

This is the average term multiplied by the number of terms. When the number of terms is odd, the average term is the middle one (term 3 when there are 5 terms).