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Question

There is a series of numbers denoted by a1, a2, a3, ... aX (Not in Arithmetic Progression!). Here's what we have:

a1 = 1
and
a(M + N) = aM + aN + MN

Using the two given data, find the 2001st term.





In case you have trouble understanding:
a1 is the first term, a2 is the second term and so on. aX is the xth term.
a(M + N) denotes the (M + N)th term of the series.

And remember that this is NOT an Arithmetic Progression!

Answer 1

a(M + N) = aM + aN + MN
Putting M = 1
=> a(N + 1) = a1 + aN + N
=> a(N + 1) - aN = N + 1
Putting N = 1,
=> a(2) - a(1) = 1 + 1
=> a(2) = 3
Putting N = 2,
=> a(3) - a2 = 3
=> a(3) = 6
Thus, series is 1, 3, 6, 10, 15, ....
Thus, a(N) = ΣN
=> a(2001)
= Σ2001
= (2001) * (2002) / 2
= 2003001.

Answer 2

The answer : a2001=2003001

a2=a(1+1) = a1+a1+1= 3
a3=a(2+1) = a2+a1+2 = 6
a6=a(3+3) = a3+a3+9 = 21
a7=a(6+1) = a6+a1+6 = 28
a14=a(7+7) = a7+a7+49 = 105
a15=a(14+1) = a14+a1+14 = 120
a30=a(15+15) = a15+a15+225 = 465
a31=a(30+1) = a30+a1+30 = 496
a62=a(31+31) = a31+a31+961 = 1953
a124=a(62+62) = a62+a62+3844 = 7750
a125=a(124+1) = a124+a1+124 = 7875
a250=a(125+125) = a125+a125+15625 = 31375
a500=a(250+250) = a250+a250+62500 = 125250
a1000=a(500+500) = a500+a500+250000 = 500500
a2000=a(1000+1000) = a1000+a1000+1000000 = 2001000
and finally
a2001=a(2000+1) = a2000+a1+2000 = 2003001

a rather brute forced method though...

Answer 3

A(M+1) = A(M) + A(1) + M*1 = A(M) + (M+1)

So A must be the triangular numbers:

A(M) = M(M+1)/2

You can see that the triangular numbers have the feature listed above for general N by drawing a triangle of dots with size M+N and then removing the MxN corner, leaving two triangles of size M and N. So (M+N+1)(M+N)/2 = M(M+1)/2 + N(N+1)/2 + MN

[ You could, of course, prove it using brute force algebraic manipulations.]

Answer 4

We can write this using n = 1 always.

Thus a(1) = 1 ; a(2) = a(1 + 1) ; a(3) = a(2 + 1) ,....

Using this we get a(2001) = 2003001.

Answer 5

This appears to be another approach to summations:

Summation(2001)=summation(2000)+2001
=2003001

Answer 6

it would be a345