Given the sum of a series and a term how would you find Tn?

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Discussion Overview

The discussion revolves around the problem of finding the nth term of a series when given the sum of the series and a specific term. It includes considerations of both arithmetic and geometric series, as well as the implications of having multiple series that can fit the same criteria.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant asserts that it is impossible to determine the nth term given only a specific term and the sum of the series, citing the existence of multiple series that can satisfy these conditions.
  • Another participant presents a solution involving an arithmetic series, providing specific values for the 6th term and the sum of the first 9 terms, and derives the formula for the nth term.
  • A later reply questions the notation used in the solution, seeking clarification on the terms involved.
  • Clarifications are made regarding the definitions of the first term, the common difference, the specific term, and the sum of the series.

Areas of Agreement / Disagreement

Participants do not reach a consensus; there are competing views on the feasibility of determining the nth term from the given information, with one participant arguing it is impossible while another provides a solution based on an arithmetic series.

Contextual Notes

There are unresolved aspects regarding the assumptions made about the type of series and the specific definitions of terms used in the discussion.

Superposed_Cat
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Hi all, given the sum of a series and a single term how would one find the nth term? Any help appreciated.
 
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You can't. That's obviously impossible. There exist an infinite number of series having a given specific term and converging to a give value.

Suppose you are given the term "a_i", for i a specific number, and that the sum is "S". Choose "A" to be any number and write the geometric sequence that converges to S- a_i. Then the series consisting first term A, the rest of the geometric series with "a_i" stuck in a the ith place, has sum S.
 
It's arithmetic, never mind I figured it out, apologies for wasted time.

t6=32
s9=234
32=a+5d
d=(32-a)/5

234=0.5*9(2a+8((32-a)/5))
a=2
d=6
Tn=6n-4
 
In your original post you said "given the sum of a series and a single term". What are "t6", "s9", "a" and "d".
 
a is the first term, d is the difference between terms, t6 is 6th term, s9 is sum of the first 9 terms
 

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