Plus, Minus, Divide

Calculus Level 2

lim n ( j = 1 n 2 j ) ( j = 1 n 1 ) 2 j = 1 n 1 = ? \large \lim_{n\to\infty} \frac{\left(\displaystyle\sum_{j=1}^n 2j \right) - \left(\displaystyle \sum_{j=1}^n 1 \right)^2}{\displaystyle \sum_{j=1}^n 1} = \ ?


The answer is 1.

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4 solutions

Siddharth Singh
Nov 22, 2015

Using the sum of the first n n natural numbers formula ,

( 2 n ( n + 1 ) 2 n ) n \left(\dfrac{2\frac{n(n+1)}{2}}{n}\right)-n

n ( n + 1 ) n n \frac{n(n+1)}{n}-n

n + 1 n = 1 n+1-n=1

What a laughter! I up voted. The question does mean that each series are having exactly same total number of terms. However, strict staff shall oppose to you I think.

Lu Chee Ket - 5 years, 6 months ago

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Yes, my initial intention was like that, but I've already changed it to a limit form to avoid infinity dilemma.

Worranat Pakornrat - 5 years, 6 months ago

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Now they share exactly the same description at once. However, many people shall find this easy because of the change of description. Hard to make it fantastic and also complying to mathematics wanted.

Lu Chee Ket - 5 years, 6 months ago

Guys!

Sum of natural numbers till 'n' = n(n+1)÷2 Here:
2+4+6+8... = 2(1+2+3+4...) So, n(n+1)÷2 = 2(1+2+3+4...) => n(n+1) = 1+2+3+4...

Now we can say that n = infinity as the series does not end. So, n=1+1+1+1... => taken alphabetically, [{n(n+1)}÷n] - n => {(n²+n)-n²}÷n = n÷n = 1

I think this is the right method to solve the question. Any corrections are welcomed.

Akhash Raja Raam - 5 years, 6 months ago

Why (1+1+1..……)=n and why there is n in(n(n+1)/2)?

Aparna Kalbande - 5 years, 6 months ago

But how? What does n stand for?

Thaddeus June - 5 years, 6 months ago
Akhil Bansal
Nov 23, 2015

= lim n ( j = 1 n 2 j ) ( j = 1 n 1 ) 2 j = 1 n 1 \large = \displaystyle \lim_{n\to\infty} \dfrac{\left(\displaystyle \sum_{j=1}^n 2j \right) - \left(\displaystyle \sum_{j=1}^n 1 \right)^2}{ \displaystyle \sum_{j=1}^n 1} = 2 n ( n + 1 ) 2 n 2 n \large = \dfrac{2\dfrac{n(n+1)}{2} - n^2}{n}

= n + 1 n = 1 \large = n + 1 - n = \boxed{1}

Xiaoying Qin
Nov 23, 2015

When you see a problem that looks extremely hard but its like 30 points say 1.

Ankit Raj
Nov 23, 2015

I don't know sign mean but its like a simple algebraic expression j=1

then 2j =2

so it will be (2-1)/1=1

Wrong .... Not clear

Syed Baqir - 5 years, 6 months ago

Yaa I don't know sign before but I know now

ankit raj - 4 years, 5 months ago

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