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Depth · one concept from first principles · step 2 of 4

Every difference quotient is the slope

A linear function f(x)f(x) satisfies

f(10)−f(1)9+f(9)−f(2)7+f(8)−f(3)5+f(7)−f(4)3+f(6)−f(5)1=10\frac{f(10)-f(1)}{9}+\frac{f(9)-f(2)}{7}+\frac{f(8)-f(3)}{5}+\frac{f(7)-f(4)}{3}+\frac{f(6)-f(5)}{1}=10

Find the value of f(50)−f(24)f(50)-f(24).

ExpressionsFunctions
Your answer
How to solve
Strategy Look for a Pattern — Look for the pattern in the denominators: 9=10-1, 7=9-2, 5=8-3, 3=7-4, 1=6-5. Each one is exactly the gap between the two inputs on top, so every fraction is (change in y)/(change in x) -- the slope. The scary sum is just five copies of the slope.
1STEP 1

Spot the pattern in the denominators

Each denominator equals the difference of the inputs in its numerator.

10-1=9, 9-2=7, 8-3=5, 7-4=3, 6-5=1
2STEP 2

Each fraction is the slope

For a linear function, (f(q)-f(p))/(q-p) is the slope m, no matter which two inputs you pick. So the sum is 5m.

m+m+m+m+m=5m=10 → m=2
3STEP 3

Use the slope on the new pair

Apply the same rule to the inputs 24 and 50.

(f(50)-f(24))/(50-24)=2 → f(50)-f(24)=2 × 26=52
Answer
52
Take a concrete example, f(x)=2x: the fractions are 18/9=2, 14/7=2, 10/5=2, 6/3=2, 2/1=2, total 10; and f(50)-f(24)=100-48=52.
Takeaway

Before you grind through a messy sum, check whether every piece is secretly the same thing -- here each one was just the slope.

  • Spot the pattern in the denominators
  • Each fraction is the slope
  • Use the slope on the new pair
Where next?
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▶ Practice — 12 problems