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Question
I've hardly give-up on mathematics whenever i remember and sight  this question: the sum of the first twenty-one terms of a linear sequence(A.P.) is 28, and the sum of the first twenty-eight terms is 21. Find which term of the sequence is  0 and also the sum of the terms preceding it.  sir, i believe that you can convert my dreppression to happiness by solving the question effectively. may God bless you, Sir.

Answer
What you call a linear sequence is also know as an arithmetic sequence which has the form

An+1 = An + d  where An+1 is the (n+1)th term of the sequence and is related to the nth term, An, by adding the fixed constant d. Note that

An = A1 + (n-1)d

The sum of the first n terms is

Sn = sum(n=1,N){An} = A1 + A2 + ... + AN.

This can also be written Sn = (n/2)(A1 + AN).

For the current problem, we have

(21/2)(A1 + A21) = (21/2)(A1 + 20d) = 28 and
(28/2)(A1 + A28) = (28/2)(A1 + 27d) = 21.

These are 2 independent  equations in 2 unknowns, namely A1 and d. A little algebra gives

A1 = 3.03
d = -0.17.

We also want to find which term in the sequence is 0 (or close to it) so

Aj = 0 = A1 + (j-1)(-0.17)  => j = 19.

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randy patton

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college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography

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26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

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J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

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M.S. MIT Physical Oceanography, B.S. UC Berkeley Applied Math

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