Download (1) A regular triangle of side n is divided uniformly into regular

Sums and Geometry
Here n > 0 is a whole number.
(1) A regular triangle of side n is divided uniformly into regular triangles of side 1. How many
such triangles are there? (can you find the exact number when n = 100?)
Hint: The sum of the first n odd numbers is n2 . Complete the large triangle to a large
parallelogram. We now have twice the initial number of triangles. But we can pair these
triangles two by two to form parallelograms. Now count the parallelograms of side 1.
(2) A regular triangle of side n is divided uniformly into regular triangles of side 1. How many
dots (vertices) are there in the picture? (can you find the exact number when n = 100?)
Hint: The sum of the first n + 1 positive integers is (n + 1)(n + 2)/2. Complete the large
triangle to a large parallelogram of sides n and n + 1, then count the dots and divide by 2.
Another way: Add up: (n + 1)2 = n2 + 2n + 1, n2 = (n − 1)2 + 2(n − 1) + 1, ..., etc.
(3) There are n + 1 houses in a village and each house is connected with any other house by
exactly one bridge. How many bridges are there in total?
n+1
Hint: There are two ways to count
bridges and they lead to the formula from
2
the previous ex.
(4) A regular triangle of side n is divided uniformly into regular triangles of side 1. How many
regular triangles of side 2 are there in the picture? (can you find the exact number when
n = 100?)
Hint: Counting the triangles with the same orientation as the original triangle, and then
those upside down: 1+2+...+(n−1)+1+2+...+(n−3) = (n−1)n
+ (n−3)(n−2)
= n2 −3n+3
2
2
for n > 1.
Alternatively, by the method applied for question (1), we complete the triangle to a
parallelogram and thus double the number of side 2 triangles, but we have also added
in those (n − 2) side 2 triangles which intersect the diagonal of the parallelogram and
are thus neither included in the original triangle nor in its ”mirror”. We get (n − 1)2
parallelograms of side 2 in a parallelogram of side n. After subtracting (n − 2) we get
(n − 1)2 − (n − 2) = n2 − 3n + 3.
(5) A regular triangle of side n is divided uniformly into regular triangles of side 1. How many
regular triangles are there in the picture?
Hint: There are n2 triangles of side 1, 1 + 2 + ... + (n − 1) + 1 + 2 + ... + (n − 3) of side
2, 1 + 2 + ... + (n − 2) + 1 + 2 + ... + (n − 5) of side 3... so
X n+1−k X n + 1 − 2k +
2
2
1≤k≤n
0<2k≤n
then also use the formula from the next ex.
(6) A square of side n is divided uniformly into squares of side 1. How many squares can you
count in the picture?
Hint: Use (n + 1)3 = n3 + 3n2 + 3n + 1 inductively to calculate 1 + 22 + ... + n2 . (Picture
of divided cube here).
1
2
(7) A cube of side n is divided uniformly into cubes of side 1. How many cubes are there in
the picture?
Hint: Use formula for (n + 1)4 inductively to calculate 1 + 23 + ... + n3 .
(8) Find a formula for 12 − 22 + 32 − ... + (2n + 1)2 − (2n + 2)2 and calculate this when n = 100.
Hint: Group them in pairs
P and use the difference of squares formula, or group them in
even and odd and expand k (2k + 1)2 .
(9) Find a short formula for 1 · 2 + 2 · 3 + 3 · 4 + ... + n(n + 1) (can you find the exact number
when n = 100?). Find a geometric proof for your formula.
Hint: n(n + 1)(n + 2)/3. Can compute based on formulas for 1 + 22 + ... + n2 and
1 + 2 + ... + n. Or, geometric picture in attached file.
(10) Ann, Ben and Cian need to choose one present each from a total of (n + 2) presents labeled
by numbers from 1 to n + 2. For example, one choice is: (Ann–1, Ben–2, Cian–3). How
many such possible choices are there? Suppose that you know that (k + 2) is the largest
label number among the three presents chosen. How many possible choices are there in this
case? Can you use this to prove the formula in the previous exercise?
Hint: this is context for the previous formula: 3(1 · 2 + 2 · 3 + 3 · 4 + ... + n(n + 1)) =
n(n + 1)(n + 2).
(11) Find a short formula for 1 · 2 · 3 + 2 · 3 · 4 + ... + n(n + 1)(n + 2).
Hint: like in previous ex. but with 4 people...
(12)
(13)
(14)
(15)
Can you generalize the above for 1 · 2 · 3 · ... · (k + 1) + ... + n(n + 1)(n + 2)...(n + k)?.
Find a formula for 1 + 24 + ... + n4 and calculate the sum when n = 100.
Use induction to prove all formulas found in the previous exercises.
Consider the table:
1
3+5
7 + 9 + 11
13 + 15 + 17 + 19
= 1
= 8
= 27
= 64.
Guess the general law suggested by these examples and prove it.
Anca Mustata, School of Mathematical Sciences, UCC
E-mail address: [email protected]