This chapter looks at where the sum formula for an infinite geometric series is actually used.
A repeating decimal such as 0.777… and the total area of shapes that keep shrinking as you join midpoints can both be computed with a/(1 − r) once you find the first term and the ratio.
Shade half of a square, then half of what is left, and keep going forever: the shaded areas add up to 1/2 + 1/4 + 1/8 + … = 1.
In the picture, as you shade more pieces the leftover area 1/2ᵏ shrinks and Sₖ moves toward 1.
Repeating Decimals as Fractions
🔁 A repeating decimal is an infinite geometric series
①0.777… can be split into 7/10 + 7/100 + 7/1000 + …
②That is an infinite geometric series with first term 7/10 and ratio 1/10
③Since |1/10| < 1 it converges, and its sum is the fraction the decimal stands for
Write the repeating decimal 0.777… as a fraction in lowest terms.
1
View it as an infinite geometric series with first term 7/10 and ratio 1/10.
2
Substitute into the sum formula S = a/(1 − r).
S = 7/101 - 1/10 = 7/109/10
▸ 7/9
A one-digit repeating block gives ratio 1/10 and a 9 in the denominator; a two-digit block gives ratio 1/100 and 99.
Converting Repeating Decimals
Repeating decimal as a fraction
0.aaa… = a/9 , 0.ababab… = (ab)/99
(ab) means the two-digit number, not a product. Write as many 9s in the denominator as the repeating block has digits; for example 0.363636… = 36/99 = 4/11
An Infinite Sum You Can See
seegongsik.com
Shading a square half at a time shows 1/2 + 1/4 + 1/8 + … approaching 1
3
The halving series
12 + 14 + 18 + ⋯ = 1/21 - 1/2 = 1
First term 1/2 and ratio 1/2, so the sum is exactly 1
🟧 Half of a half of a half
①Shade half of a square of area 1 and you have 1/2
②Shading half of what remains adds 1/4 and leaves 1/4 unshaded
③After k pieces the leftover area is always 1/2ᵏ, so the shaded total is Sₖ = 1 − 1/2ᵏ
④As k grows without bound the leftover area approaches 0, so the sum converges to the whole square, which is 1
Total Area of Similar Shapes
Example 2
Join the midpoints of the sides of a square with side 2 to make a new square, and repeat forever. Find the total area of all the squares.
1
The midpoint square has a side 1/√2 times the previous one, so its area is half as large. The ratio is 1/2.
2
The first square has area 4, so S = 4/(1 − 1/2).
S = 41/2 = 8
▸ 8
If the similarity ratio is k, the ratio of areas is k². When the question asks for a total area, take the ratio of areas as the common ratio.
Summary
The key to applications
|r|<1 ⇒ S = a1 - r
For repeating decimals, areas of shapes and sums of lengths alike, find the first term a and the ratio r and substitute
2021 provincial mock exam Math type, adapted
Join the midpoints of the sides of square S₁, whose side is 6, to make square S₂, and repeat forever. What is the total area of all the squares?
①48
②60
③72
④96
⑤Diverges
▸ ③ 72
1
The midpoint square has half the area of the previous square (ratio 1/2).
2
S₁ has area 36, so S = 36/(1 − 1/2).
S = 361/2 = 72
🎯 Exam Points
①A repeating decimal is an infinite geometric series with ratio 1/10 or 1/100
②The number of digits in the repeating block is the number of 9s in the denominator
③In shape problems, a similarity ratio k gives an area ratio k² and a length ratio k
④Take the first term exactly from the first shape
⑤Check |r| < 1, then substitute into S = a/(1 − r)