Tricky Triangle

Number Theory Level 2

What is the largest triangle number that is less than 1000?

Details and assumptions

A triangle number has the form $\frac {n(n+1)}{2}$ for some positive integer $n$ .

The answer is 990.

3 solutions

Shubhrajit Santra
Aug 10, 2013

According to the question, n(n+1)/2<1000=> n^2+n<2000=> n^2+n-2000<0 On solving the quadratic equation we get: n= -91/2 or 89/2. But, negative answer is not possible, therefore -91/2 is rejected and 89/2 or 44.5 is accepted. However 'n' cannot be a fractional number, hence the value of 'n' is 44. (It is not 45 since it will make the triangle number larger than 1000). Therefore, taking n = 45, the largest triangle number that is less than 1000 is 990.

thank you so much, I thought I did mistake because I forgot and wrote in the answer the number 44 not 990 it's correct, but you wrote a mistake, in the last line n=45, it is n=44

Yassir El Attar - 7 years, 10 months ago

Stefanus Arya
Aug 9, 2013

We must use this form n(n+1)/2. ;if n = 100 then 100(100+1)/2 = 5050. #More than 1000!! ;if n = 50 then 50(50+1)/2 = 1275 #Near to 1000. ;if n = 45 then 45(45+1)/2 = 1035 #Oh! ;if n = 44 then 44(44+1)/2 = 990 #So the answer is 990.

Samagra Sharma
Aug 5, 2013

We have {n(n+1)}/2 <1000 n(n+1) < 2000 n(n+1)-2000<0 n<45 Therefore putting n=44 in {n(n+1)}/2 we get 990

How did you go from n(n+1) - 2000 < 0 to n < 45? That's the only step I don't understand...

Richard Steele - 7 years, 10 months ago

by solving the quadratic inequality n^2 + n - 2000 < 0 and finding the greatest integer value possible

Samagra Sharma - 7 years, 10 months ago

awesome

Manish Kumar - 7 years, 10 months ago

kuch samja hi nahi......

Prk Jain - 7 years, 10 months ago

Tricky Triangle

The answer is 990.

3 solutions

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