Find how many numbers lie between two consecutive perfect squares. Do you notice a pattern?
Answer
Between n2 and (n+1)2 there are exactly 2n numbers.
Squares
Numbers strictly between
How many
2n
1 and 4
2, 3
2
2 × 1
4 and 9
5, 6, 7, 8
4
2 × 2
9 and 16
10 … 15
6
2 × 3
16 and 25
17 … 24
8
2 × 4
Why it happens: The gap itself is (n + 1)2 – n2 = 2n + 1. That count includes one of the two end points, so the numbers strictly in between number 2n + 1 – 1 = 2n. Notice the answer is always even, so no two consecutive squares are ever "close together" once n grows.
Tip: The official answer writes this as q – p – 1 for consecutive squares p and q. That is the same thing: with p = n2 and q = (n+1)2, q – p – 1 = (2n + 1) – 1 = 2n.
Q2.
How many square numbers are there between 1 and 100? How many are between 101 and 200? Using the table of squares you filled earlier, enter the values below, tabulating the number of squares in each block of 100. What is the largest square less than 1000?
Answer
Blockwise count of the perfect squares up to 1000:
Block
Squares in it
How many
1 – 100
1, 4, 9, 16, 25, 36, 49, 64, 81, 100
10
101 – 200
121, 144, 169, 196
4
201 – 300
225, 256, 289
3
301 – 400
324, 361, 400
3
401 – 500
441, 484
2
501 – 600
529, 576
2
601 – 700
625, 676
2
701 – 800
729, 784
2
801 – 900
841, 900
2
901 – 1000
961
1
Total = 10 + 4 + 3 + 3 + 2 + 2 + 2 + 2 + 2 + 1 = 31 squares below 1000, and the largest is
312 = 961 (the next square, 322 = 1024, is past 1000)
Why the counts thin out: The gaps between consecutive squares are 2n + 1, which keeps growing. Early on the gaps are 3, 5, 7 — many squares fit in a block of 100. Around 900 the gap is already about 60, so only one or two squares can fall in each block.
Q3.
Do you remember triangular numbers? Can you see any relation between triangular numbers and square numbers? Extend the pattern shown and draw the next term.
Answer
Yes — two consecutive triangular numbers always add to a square.
So the next picture is a 5 × 5 square of dots split by a staircase into a triangle of 10 dots and a triangle of 15 dots.
A 5 × 5 square of dots cut into the triangular numbers 10 and 15.
Why it happens: The n-th triangular number is Tn = 1 + 2 + … + n = n(n+1)/2. Then Tn–1 + Tn = (n–1)n/2 + n(n+1)/2 = n[(n–1) + (n+1)]/2 = n × 2n/2 = n2. Geometrically, two staircase triangles of heights n – 1 and n slot together into an n × n square.
Q4.
The area of a square is 49 sq. cm. What is the length of its side?
Answer
The side is the number that gives 49 when multiplied by itself.
side × side = 49 7 × 7 = 49, so side = 7 cm
We call 7 the square root of 49, and write √49 = 7.
Why we take only 7: Algebraically both 7 and –7 square to 49, so 49 has two integer square roots. A side length is a measured length, so only the positive root makes sense here. That is why this chapter works with the positive square root throughout.