Could you please tell me how to find the summation of a series whose difference of difference (2nd order difference) of successive terms is a constant.
Ex- 3+4+9+18+31+...... 1st order difference is 4-3=1
9-4=5
18-9=9. 31-18=13 Second order difference is 5-1=4, 9-5=4, 13-9=4
Hence second order difference is a constant.so how do you find the summation of such a series upto n terms.
Also it would be helpful if you could explain a general procedure if the nth order difference was a constant.
Thanks in advance.
P.s would be grateful to anyone who would reply at the earliest as I need it for an exam the day after tomorrow
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I think the nth term of these sequences (constant second order difference) can be expressed in the form an2+bn+c, you can then find a,b,c by substituting in values of n: For example with your example a+b+c=3,4a+2b+c=4,9a+3b+c=9, solving yields a=2,b=−5,c=6, so the nth term is 2n2−5n+6, you can then use the sum of squares formula 6n(n+1)(2n+1) and the sum of first n positive integers formula 2n(n+1) to evaluate the sum.
I suspect (but currently have no proof) that if the kth order difference is constant, then each member of the series is a polynomial of order k, so you could get the sum by using formulae for the sum of the kth, k−1th...1st powers of the first n positive integers.
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Definitely, your suspicion is correct since for a kth order polynomial you can create a finite difference table whose kth row is constant...
Thanks.but could you elaborate as to how you deduced that it must of the form an^2+bn+c.
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you can think of it as a function whose second derivative is constant, hence the polynomial for the sequence will be a quadratic.
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SEE..here we see there are two series..one is the one that you gave..another is the series of the differences..let us call the series you gave {a}n..(subscript n) and the series of the differences 1,5,9,13,.. as {b}n...now if you notice
{a}n - {a}n-1 = {b}n-1
i.e. the difference of the nth term and the n-1 th term of the series in the question is equal to the n-1 th term of the series of the differences (1,3,5,.. i.e. {b}n)
therefore,
_________ [summing them all up] {a}n - {a}1 = {b}n-1 + {b}n-2 + ..... + {b}1 .....................(eqn 1)
now {b}n is a general arithmetic progression series..we can find the general term of {b}n {b}n = 4n-3 find summation of {b}n, where n varies from n-1 to 1
so we get to know the right hand side of equation 1 {a}1=3 [given] therefore find {a}n
it'll be {a}n=2n^2 - 5n + 6
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Thanks a lot that was very neatly explained.
pranav, the formula for the summation of this series is, n.a +(∑(n-1))a '+(∑(n-2))d where, a is the first number of the series a' is the first number of the secondary series which is obtained by 1st order difference d is the constant which is obtained by the 2nd order difference a is 3 , a' is 1 , d is 4 in your given example note this formula is only applicable where n is greater than 1 this is my 1st post, hope it is ueful for u
how to simple continued fraction of 41/29
no -_-