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# what are the square numbers between 300 and 500 and also what are the two triangular numbers between 50 and 70

PLASE HELP!

### 6 Answers

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• Favourite answer

Let's start with the more interesting first...

Triangular numbers are those that represent the number of objects in an evenly spaced triangle.

1

3

6

The formula is easy to see.. the first row has only 1 object

The second row has 2 objects

The third row has 3 objects

The fourth row has 4 objects...

Get the idea? To figure out how many objects are in the 8th row, take

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36

Trial and error tells us that

10 -> 36 + 9 + 10 = 55

11 -> 55 + 11 = 66

Now for the square numbers between 300 and 500.

Let's figure out where to start.

sqrt(300) = 17.32

sqrt(500) = 22.36

So all the numbers between 18 and 22 will work!

18² = 324

19² = 361

20² = .......

• Since the nature of triangular numbers have been given, I give you the formula for the nth triangular number (let it be t_n)

t_n = n(n + 1)/2

Now, we set up an equation. Since we want to find triangular numbers between 50 and 70, then

50 < t_n < 70

We substitute the formula for t_n

50 < n(n + 1)/2 < 70

We multiply all members of the inequality by 2, to cancel the denominator in the middle.

100 < n(n + 1) < 140

Since n and n + 1 are consecutive integers, then we want to find 2 consecutive integers whose products are between 100 and 140. Since 10 x 11 = 110 and 11 x 12 = 132, then 10 and 11 are what we are looking for.

Thus, the triangular numbers are t_10 and t_11, or

t_10 = 10(10 + 1)/2 = 10(11)/2 = 5(11) = 55

t_11 = 11(11 + 1)/2 = 11(12)/2 = 11(6) = 66

Therefore, the two triangular numbers between 50 and 70 are 55 and 66.

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A square is an integer which is an integer multiplied by itself. Let s be the square number. Thus, 4 is a square (since 2 · 2 = 4 and -2 · -2 = 4) and 100 is a square (since 10 · 10 = 100 and -10 · -10 = 100)

Thus, the squares between 300 and 500 are 324, 361, 400, 441, and 484. (18·18, 19·19, 20·20, 21·21, and 22·22, respectively)

^_^_^

• Hi there.

Firstly, the square roots of 300 & 500 are approximately 17.32 and 22.36 respectively. Therefore, the squares in this range have roots between these limits - i.e. 18, 19, 20, 21 and 22. The squares in your target range are thus 324, 361, 400, 441 and 484.

Then we have the triangular numbers which actually have a similar format. The idea of a trianglar number is that it is the sum of a full sequence of numbers from 1 to whatever, this gives the format of y=(x^2 + x)/2 where x represents the upper limit. Since this is clearly a quadratic equation with a standard formula for solving, no trial-and-error session is required. Apply the standard formula and you find the roots of your limits of 50 and 70 are 9.512 and 11.343 respectively.

Within these limits, only two integers exist - these being 10 and 11. The two triangular numbers are thus 55 and 66.

I hope this helps.

Source(s): Pure mathematics and an expertise in mental calculations which has brought me the Grandmaster title and three world championships
• Anonymous
1 decade ago

There are five square numbers between 300 and 500: 324 (18^2), 361 (19^2), 400 (20^2), 441 (21^2), and 484 (22^2). There are two triangular numbers between 50 and 70: 55 (1+2+3+4+5+6+7+8+9+10) and 66 (1+2+3+4+5+6+7+8+9+10+11). Check out http://en.wikipedia.org/wiki/Square_number and http://en.wikipedia.org/wiki/Triangular_number for good basic rundowns of both types of numbers.

• Anonymous
1 decade ago

18 squared = 324 19 squared = 361 20 squared = 400 21 squared = 441 22 squared = 484

triangular numbers 55 and 66

• Here is a very trivial solution. Between 300 and 500 there will be two type of numbers : 3xy and 4xy. Now if 3xy satisfies the condition (i.e. it has a single 4) then 4xy won't (it will have double 4). If 3xy doesn't satisfies the condition(i.e. it doesn't contain any 4) then 4xy will satisfy it. So each time either of 3xy or 4xy will satisfy the condition with only exception being 344 and 444. So in total there will be 100 (for either 3xy or 4xy) - 1 (for neither 344 nor 444) = 99 such numbers. Ans = 99

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