Consecutive towers are built, as shown in the figure above.
The
1
st
tower has one floor made of two cards.
The
2
nd
tower has two floors made of seven cards.
The
3
rd
tower has three floors made of fifteen cards, and so on.
How many cards will the 1 0 0 0 th tower have?
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Nice observation (+1)
Relevant wiki: Arithmetic Progressions
In figure 1, there are
2
cards.
In figure 2, there are
7
cards.
In figure 3, there are
1
5
cards.
(For better proof):-
In figure 4, there will be
2
6
cards.
In figure 5, there will be
4
0
cards.
So, the number of cards in each case is increasing by the arithmetic progression
5
,
8
,
1
1
,
⋯
.
Now, the second figure has
5
cards more than the first term. The third figure has
8
more cards than second figure and so on. So, we see that the number of cards in the
n
th
figure exceeds the number of cards in the
(
n
−
1
)
th
by the
(
n
−
1
)
th
term of the above stated AP. So, the number of cards in the
1
0
0
0
th
figure exceeds that of the
9
9
9
th
figure by the
9
9
9
th
term of the AP.
So, the number of cards in the 1 0 0 0 th is 2 + sum of first 999 terms of the AP = 2 + 2 9 9 9 ( 2 × 5 + ( 9 9 9 − 1 ) × 3 ) = 2 + 2 9 9 9 ( 1 0 + 9 9 8 × 3 ) = 2 + 2 9 9 9 ( 1 0 + 2 9 9 4 ) = 2 + 2 9 9 9 × 3 0 0 4 = 2 + ( 9 9 9 × 1 5 0 2 ) = 2 + 1 5 0 0 4 9 8 = 1 5 0 0 5 0 0
Nice solution! These numbers are known as the "second pentagonal" numbers.
Another equivalent definition for them is the sum of the first n consecutive integers greater than n , for example:
2
3 + 4
4 + 5 + 6
etc.
and the general formula is 2 n ( 3 n + 1 ) .
Wow, using progressions to solve this!
Did the same way. :D
1st 2nd 3rd 2 7 15 and so on... (a geometric sequence)
Double the terms by a factor of 2 to find more factors
4, 14, 30...
4: 2 2, 4 1 14: 2 7, 14 1 30: 3 10, 5 6, 15 2 (look for sequences in the factor pairs of the values.) 4: 4 1 14: 2 7 30: 3 10
Sequence of 1,2,3 is just "n" Sequence of 4,7,10 is just 3n+1 Find the value by multiplying both, since they are just factors that share sequences. n(3n+1)=3n^2+n But since we multiplied by 2 to find more factors, divide sequence by 2 2 3 n 2 + n Plug 1000 into the sequence, and you get 1500500.
1st tower=2 cards 2nd tower=7 cards 3rd tower=15 cards 4th tower=26 cards
The difference between 1st and 2nd tower=5 cards The difference between 2nd and 3rd tower=8 cards The difference between 3rd and 4th tower=11 cards
Each time, the difference of the cards will go up 3
Therefore the formula is the nth tower has 1.5n^2 + 0.5n cards
Therefore the 1000th tower has 1.5(1000^2) + 500=1500500
Observe the number of cards coming from the teepees and the number of cards coming from the floors.
From the teepee pieces:
t(1) = 2
t(2) = 2+4
t(3) = 2+3+6
...
t(n) = 2(1+2+3+...+(n-2)+(n-1)+n) = 2( 2 n ( n + 1 ) ) = n(n+1)
From the floor pieces:
f(1) = 0
f(2) = 1
f(3) = 1+2
f(4) = 1+2+3
...
f(n) = (1+2+3+...+(n-2)+(n-1)) = 2 ( n − 1 ) ( n − 1 + 1 ) = 2 n ( n − 1 )
Total:
S(n) = t(n) + f(n) = n(n+1) + 2 n ( n − 1 ) = n( 2 3 n + 1 )
S(1000) = 1000( 2 3 0 0 1 ) = 1,500,500
Alternatively, you can think of the pyramids as rows of triangles with the bottom floor missing.
S(1) = 1x3 - 1
S(2) = 1x3 + 2x3 -2
S(3) = 1x3 + 2x3 + 3x3 - 3
...
S(n) = 3( 1 + 2 + 3 + ... + n ) - n = 3((n(n+1))/2) - n = n((3n + 1)/2), which is the same solution as above.
This is what I did:
First I saw a pattern
2 ∗ step number + ( step number − 1 ) + answer from last step
Then I wrote a program (In Python 3):
i=1 #Step number
x=0 #Number of cards
n=1001 #Max steps + 1
while i <= n:
if i < n:
x = 2*i + i-1 + x
i = i + 1
elif i == n:
print(x)
break
Result:
1 5 0 0 5 0 0
Notice the pattern, 1st system- 2 cards, 2nd system 4+2+1 cards,3rd system 6+4+2+3 cards.Also note that 1,3,...... are pyramid numbers. The 1000th system consists of - (2+4+6+....to the 1000-th term)+1000(1000+1)÷2=1500500 cards
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Notice that for the n th floor of the tower, you have:
2 n slanted cards and ( n − 1 ) horizontal cards.
Eg. From the 3 rd tower:
The 1 st row has 2 slanted cards and 0 horizontal cards
The 2 nd row has 4 slanted cards and 1 horizontal card
The 3 rd row has 6 slanted cards and 2 horizontal cards
Therefore, we can say that the number of cards in the n th tower, c n is given by:
c n = p = 1 ∑ n [ ( 2 p ) + ( p − 1 ) ] = p = 1 ∑ n ( 3 p − 1 )
Therefore, the number of cards in the 1 0 0 0 th tower:
c 1 0 0 0 = p = 1 ∑ 1 0 0 0 ( 3 p − 1 ) = 3 p = 1 ∑ 1 0 0 0 p − p = 1 ∑ 1 0 0 0 1 = 3 2 1 0 0 0 ( 1 0 0 1 ) − 1 0 0 0 = 3 0 0 3 ( 5 0 0 ) − 1 0 0 0 = 1 5 0 1 5 0 0 − 1 0 0 0 = 1 5 0 0 5 0 0