SOLUTION: This question is really confusing, my tutor ended up answering it in class but everyone was still kinda dazed when we left class. Will you please try to explain or solve this in a

Question 30794: This question is really confusing, my tutor ended up answering it in class but everyone was still kinda dazed when we left class. Will you please try to explain or solve this in a manner that is within a students grasp.
Question:
When the first n positive integers are added together, their sum is given by:
1/2n(n+1)
a) Demonstrate that this result holds for the case n=5.
b) Find the value of n for which the sum is 105.
c) What is the smallest value of n for which the sum exceeds 1000?
The chapter we are working on is based on factoring. Please help.
Found 2 solutions by Fermat, josmiceli:
Answer by Fermat(136) (Show Source): You can put this solution on YOUR website!
Hi there, I'n not too sure what exactly the confusion might be, so I'll just go through the problems and hope that helps. If you are still having problems, then you can relist this question, with additional points you would like to be addressed, or you can email me for info. I'm not absolutely sure how you can email me, but I thnk you can do that when you view this answer. (Look around the web page)
Sn = (1/2)n(n+1)
a) We have to show that the above formula holds for n=5.
If n is 5, then we want to add up the 1st 5 natural numbers, like so
S5 = 1 + 2 + 3 + 4 + 5
S5 = 3 + 7 + 5 (adding two numbers at a time)
S5 = 10 + 5
S5 = 15
=======
If we now use the formula for Sn, with n = 5, then we get
S5 = (1/2)5(6)
S5 = (1/2)(30)
S5 = 15
=======
So, the formula holds for n = 5.
================================
b) Find the value of n for which the sum is 105.
If Sn = 105, the we can write out an equation,
Sn = (1/2)n(n+1) = 105
(1/2)n(n+1) = 105
n(n+1) = 210 (multiplying both sides by 2)
n� + n = 210 ( expanding the lhs)
n� + n - 210 = 0 ( rearranging the eqn)
(n + 15)(n - 14) = 0 (factoring the terms)
.: n = -15, n = 14
==================
Although n = -15 satisfies the quadratic eqn, it is not a valid solution since it is negative and hence not a natural number.
Ans: n = 14
===========
c) What is the smallest value of n for which the sum exceeds 1000?
This is similar to part b), but is an inequality.
This time we can write,
Sn > 1000
substituting the formula for Sn,
(1/2)n(n+1) > 1000
n(n+1) > 2000
n� + n > 2000
Now you can go all mathematical here and make up a quadratic eqn, viz n� + n - 2000 = 0 and then use the quadratic formula to find roots of this equation and then test those roots in the original inequality to find out which one works,
or,
you can take a guess!
n� is bigger than n, much bigger if n is any decent size. So for a rough approximation, let
n� > 2000
So what's the square root of 2000?
Using my trusty calculator, I get 44.72
Since you have to add on n to the n� bit then try, as a first guess, n=44.
44� + 44 = 1980 (too small, so try the next number up)
45� + 45 = 2070 - bingo!
Ans: n= 45 is smallest number such that Sn > 1000
=================================================

Answer by josmiceli(19441) (Show Source): You can put this solution on YOUR website!
a)Demonstrate that this result holds for the case n=5.

(can do in your head or with calculator)
--------------------------
b)Find the value of n for which the sum is 105.

multiply both sides by 2

expand

use binomial formula

on
a = 1
b = 1
c = -210

(I checked this with calculator)
--------------------
c)
What is the smallest value of n for which the sum exceeds 1000?
let S = sum

use binomial formula

on
a = 1
b = 1
c = -2*S

square both sides

I want S to be a number just over 1000
That would make just over 8000
that means would be just over which is
about 89.4

trying n = 44, I get

that's a little low
try n = 45

this looks right. There may be a better way to do it- I just came up with this

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