SOLUTION: If the sum of n terms of a series is 2n^2+3n+4. Find the series and also its nth term.

Algebra ->  Sequences-and-series -> SOLUTION: If the sum of n terms of a series is 2n^2+3n+4. Find the series and also its nth term.      Log On


   

Question 1012112: If the sum of n terms of a series is 2n^2+3n+4. Find the series and also its nth term.
Answer by ValorousDawn(53) About Me  (Show Source):
You can put this solution on YOUR website!
If f(n) is a function that gives you the sum of the first n terms, the nth term is given by subtracting away the previous terms. Since terms are integers apart (1,2,3), we can extract the nth term by taking the sum to n, and subtracting away the sum of n-1. Thus, n=f(n)-f(n-1).
Solved by pluggable solver: EXPLAIN simplification of an expression
Your Result:


YOUR ANSWER


  • Graphical form: 2n%5E2%2B3n%2B4-2%2A%28n-1%29%5E2%2B3%2A%28n-1%29%2B4 simplifies to 2%2An%5E2%2B6%2An-2%2A%28n-1%29%5E2%2B5
  • Text form: 2n^2+3n+4-2*(n-1)^2+3*(n-1)+4 simplifies to 2*n^2+6*n-2*(n-1)^2+5
  • Cartoon (animation) form: simplify_cartoon%28+2n%5E2%2B3n%2B4-2%2A%28n-1%29%5E2%2B3%2A%28n-1%29%2B4+%29
    For tutors: simplify_cartoon( 2n^2+3n+4-2*(n-1)^2+3*(n-1)+4 )
  • If you have a website, here's a link to this solution.

DETAILED EXPLANATION

Look at .
Added fractions or integers together
It becomes 2%2An%5E2%2B3%2An%2Bhighlight_green%28+8+%29-2%2A%28n-1%29%5E2%2B3%2A%28n-1%29.
Look at 2%2An%5E2%2B3%2An%2Bhighlight_red%28+8+%29-2%2A%28n-1%29%5E2%2B3%2A%28n-1%29.
Moved 8 to the right of expression
It becomes 2%2An%5E2%2B3%2An-2%2A%28n-1%29%5E2%2B3%2A%28n-1%29%2Bhighlight_green%28+8+%29.
Look at 2%2An%5E2%2B3%2An-2%2A%28n-1%29%5E2%2Bhighlight_red%28+3%2A%28n-1%29+%29%2B8.
Expanded term 3 by using associative property on %28n-1%29
It becomes .
Look at .
Multiplied numerator integers
It becomes 2%2An%5E2%2B3%2An-2%2A%28n-1%29%5E2%2B3%2An-highlight_green%28+3+%29%2B8.
Look at 2%2An%5E2%2B3%2An-2%2A%28n-1%29%5E2%2B3%2An-highlight_red%28+3+%29%2Bhighlight_red%28+8+%29.
Added fractions or integers together
It becomes 2%2An%5E2%2B3%2An-2%2A%28n-1%29%5E2%2B3%2An%2Bhighlight_green%28+5+%29.
Look at 2%2An%5E2%2Bhighlight_red%28+3%2An+%29-2%2A%28n-1%29%5E2%2Bhighlight_red%28+3%2An+%29%2B5.
Eliminated similar terms highlight_red%28+3%2An+%29,highlight_red%28+3%2An+%29 replacing them with highlight_green%28+%283%2B3%29%2An+%29
It becomes 2%2An%5E2%2Bhighlight_green%28+%283%2B3%29%2An+%29-2%2A%28n-1%29%5E2%2B5.
Look at 2%2An%5E2%2B%28highlight_red%28+3+%29%2Bhighlight_red%28+3+%29%29%2An-2%2A%28n-1%29%5E2%2B5.
Added fractions or integers together
It becomes 2%2An%5E2%2B%28highlight_green%28+6+%29%29%2An-2%2A%28n-1%29%5E2%2B5.
Look at 2%2An%5E2%2Bhighlight_red%28+%28highlight_red%28+6+%29%29%2An+%29-2%2A%28n-1%29%5E2%2B5.
Remove unneeded parentheses around factor highlight_red%28+6+%29
It becomes 2%2An%5E2%2Bhighlight_green%28+6+%29%2An-2%2A%28n-1%29%5E2%2B5.
Result: 2%2An%5E2%2B6%2An-2%2A%28n-1%29%5E2%2B5

Universal Simplifier and Solver


Done!