$\large 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, \ldots$

What is the 1000th term in the sequence above?

The answer is 45.

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how do we know the no"s 44&45

Raghavender Boini
- 6 years, 7 months ago

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I think that he calculated this function 0.5n(n+1) = 1000 and he found that n is 44,... because n is natural number, so n is 44 After that, he did like his answer

Quang Trung Nguyễn
- 6 years, 7 months ago

how did u get the formula n/2(n+1)??

Manish Kumar
- 6 years, 7 months ago

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Johann Carl Friedrich Gauss

Cleres Cupertino
- 5 years, 4 months ago

Every series has a diffrent formula, i just deduced it.

Abdulrahman El Shafei
- 6 years, 6 months ago

Task-Deducing expression n(n+1)/2 for the series 1+2+3+4... Procedure-Take a square of side n unit.Fill maximum no. of square with unit length, unit breadth and unit height.Now shade n no. of squares at the base, n-1 no. of squares in the 2nd row from the base.Continue like this till all rows are shaded in this way. Observation-Now you shall see the pattern 1+2+3+4... appearing. Reasoning-If we find the shaded area it is equivalent to finding what is 1+2+3+4... .To do this- Step 1-remove n no. of squares.Step2- observing that the remaining shaded portion is the half of a rectangle we can get the no. of shaded squares dividing the area of rectangle by 2,that is length(breadth)/2 or n(n-1)/2. Step 3-adding base squares(n no. of squares) we get expression n+n(n-1)/2. Simplifying it we get n(n+1)/2.

Joseph Amal C X
- 6 years, 1 month ago

is this is median or anything else

Mujeeb Yusufzai
- 6 years, 7 months ago

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KC Yao
- 2 years, 5 months ago

he just came up with numbers that roughly squared are less than 2000 (since you take half) then since that equals 990 the next 45 added on would be 991 thru 1035

kathleen Kilmer
- 2 years, 2 months ago

I used (n^2 + 1) /2 - based on adding another upside triangle beside the first to make a parallelogram and figuring the area of that and dividing by two.

The square root of 1999 = 44.7 something, so that means the 44's are done and we are into the 45's.

Diane Sollazzo
- 2 years ago

This makes sense !!! thanks for the help !!

cute panda panda
- 6 months, 4 weeks ago

solution not clear !!!!

Ashish Tiwari
- 6 years, 7 months ago

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What is missing then? 0.5n(n+1) is just the standard formula for summing n natural numbers. 44 has been found by trial and error or using abc-formula.

Peter van der Linden
- 4 years, 7 months ago

i simply start adding the terms on calculator

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hahha yup, easy

Khyam Arain
- 6 years ago

use the formula! n(n+1)/2, it is something you should know.

Razzi Masroor
- 4 years, 3 months ago

1

22

333

4444

55555

666666 on this pattern .(n(n+1))/2) . according to this law for the value of n=10, we find the the value of term is55. where 10 is the last term of series 10. for n=44 we see the value of full term is 990.where 44 is the last term of series no 44. so definitely 45 is the 1000term of next.that means series of 45.

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Ceiling function of square root of (2000-1)

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You can use integrals.

Solve this equation for b to find the x value in which the area under the graph is 1000. This works because the area under the graph=number of terms used and x values=the term itself. I used maple because I am lazy:

Of course, the pattern doesn't allow negative values so we will ignore that. WIth the area under the graph=1000 in the x value 44.7, we can just round up to find the correct value, which is 45.

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2-repeats constantly till term 3(1+2)

3
3
3-repeats constantly till term 6(1+2+3)
4
4
4
4-repeats constantly till term 10(1+2+3+4)
...
Here observe that if a no. x is of the form 1+2+3+4+5...+n=x, then n repeats n times constantly till the term x.Think that x is a no. just below 1000.Then n repeats itself constantly till term x.
Using equation n(n+1)/2=1000 we see n=44 and x=990.Now it is easy to note that 45 starts repeating after the 990th term and that the 1000th term is 45.

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I just visualised it as a triangle with an area of 1000. The square with the same length sides would have an area of 2000. Square root of 2000 is 45 to the next whole number.

Peter Fenwick
- 5 years, 4 months ago

Absolutely brilliantly described and explained! Thanks!

Santiago Luna
- 1 year, 9 months ago

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Just Use the formula of natural number n(n+1)/2..

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In the sequence, the number 1 has 1 time, number 2 has 2 times, number 3 has 3 times, number 'n' has 'n' times. So 'n' is from (((n-1)*n)/2)+1 to (n(n+1)/2 terms...

If we took (((n-1)*n)/2)+1 = 1000

then, n = 45.something

So 1000 term is in the consecutive line of 45

So the answer is 45...

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Hi I'm trying to get better at math but I'm pretty hopeless, could you explain in simple terms how you derived the (((n-1)*n)/2)+1 = 1000?

thx much :)

Robert Speer
- 3 years, 5 months ago

Bcoz numbers r repeated according to their value i.e 1 is 1 times 2 is 2 times , 3 is 3 times , so to find thousandth term we need to find that term of series where the sum becomes approximately equal to 1000

so by sum of n terms formula in arithmetic progression 1000~ n/2(2a+(n-1)*1) or simply 1000= n(n+1)/2 or n~45 nd 44 approx no since series is ongoing so 45 will eb repeated 45 times nd in that sum 1000 would hav been completed so 45 is the right answer

thank you

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can u elaborate why we need to need to find the term of series where sum becomes approx equal to 1000 ???

Sunirudh Gujral
- 5 years, 4 months ago

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1,2,2,3,3,3,4,4,4,4...----------(a) this series can written as One+Two+Three+Four+Five.....----(b) ("Two" means in equation (b) is 2 has came 2 times in equation ( a). similarly "three" means in equation (b) is 3 has come 3 times in equation ( a) so we want 1000 term it means sn=1000 in modified equation(b) means 1000 term in equation (a)

Sn=1000 a=1 d=1 n=??? ( from equation b) (n/2)*(2a(n-1)d, after simplification n^2=2000 n=44.72~~45 so n=45

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There are 1 value with the value of 1 There are 2 values with the value of 2 ETC... 2's are from terms 2 - 3 all the numbers before the first 3 is 1, 2, 2, so 3 starts after 1+2 5's are from terms 11 - 15 all the #s before 6 is 1,2,2,3,3,3,4,4,4,4,5,5,5,5,5, so 6 strats after 1+2+3+4+5 or 15 terms.

If you add all the integers up to 45, you would get pass 1000 or 1035 to be exact. the value 45 stops at the 1035th term

If only add up to 44, you get 990.And the value 44 stops at the 990th term 1000 is between 990 and 1035 Therefore the 1000th term is 45, (because it's over 990)

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therefore we have the inequation n(n+1)/2<1000
find the largest possible value for n and then add 1

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[n(n+1)]/2

Putting up the values, you will get the answer. Obviously approx.

=(45*46)/2

=2070/2

=1035

We get 1035 which means that 45 will go on till 1035th term and the asked term is 1000.

Therefore, the required answer is 45.

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lola lola lola lola lola

A Former Brilliant Member
- 4 years, 10 months ago

$\frac{n(n+1)}{2}=1000$

$\boxed{n=45}$

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Without having the nature of the sequence explicitly stated we must make an assumption about its behavior. It seems reasonable to assume that the sequence is made up of one 1, two 2's, three 3's, and so on. If the sequence continued up to the last n, there would be $1 + 2 + ... + n$ = $\frac{1}{2}n(n+1)$ terms.