Find the sum of the First $50$ Natural Numbers starting from $11$. Is it from $11-50$ or $11-60$?

Question

This a simple question yet confusing for me, I found the answer as 1220 by taking sum from $11$ to $50$, by inferring the question as first 50 natural numbers {$1,2,3,4,5,6,7,8,9,10,11,...49,50$}

and sum starting from 11 which means sum $= 11+12+...+49+50$. But my friend says otherwise, he took from 11 to 60 and says answer as 1775. Can anyone explain how to infer the word first in the context of maths.

Answer 1

I think there's a bit of ambiguity in the language, so either interpretation is justified. I also think it's just a poorly worded question. The only recourse is to ask whoever set the question or to just let it go - I don't think the intended meaning is really something you can figure out by yourself.

To clarify why I believe both interpretations are valid:

1. "The first $50$ natural numbers" means $1-50$. So then "The first $50$ natural numbers (starting from $11$)" could mean the same set $1-50$ but starting from $11$ and excluding $1-10$. So $11-50$.

2. "The natural numbers starting from $11$" means $11,12,\ldots$. So then "(The first $50$) natural numbers starting from $11$" could mean the first $50$ elements of this set. So $11-60$.

Answer 2

There's a high school formula for the sum of consecutive terms in an arithmetic sequence: it is equal to the number of terms, times the arithmetic mean of the first and last term.

Here the first term is $11$ and $50$th term is $60$, so the sum is $$50\times\frac{11+60}2=1775.$$

Answer 3

From a book for GRE practice

I can see how the word "first" is used in a particular context in mathematics. Is that why my interpretation is wrong... or is that my interpretation does not bring logic even by english, that's what i would like to know.. thanks @jam now i know i am not alone in this. Take a look at the image link attached here, if we see the options everyone will surely come to conclusion my friend is right.. but i wanted to clarify if there is any logical mistake in my interpretation.

Answer 4

Following the link that you provided, we find that the wording in the review book is

Find the sum of first $50$ natural numbers starting from $11$.

This is not Standard English; it should say "of the first" rather than "of first". Hence I have some doubts that you will ever see a question worded exactly like this on an actual GRE exam.

But if the phrase "the first $50$ natural numbers starting from $11$" were to occur on an actual GRE, I probably would interpret it by considering the "natural numbers starting from $11$" (which are $\{11, 12, 13, \ldots\}$) and then taking the first $50$ of those.

That is, I would take the first $50$ members of $\{n \in \mathbb N \mid n \geq 11\}.$

The alternative, if I first collect the first $50$ natural numbers and then "[start] from $11$", is that the "starting from $11$" part becomes meaningless. I already have a finite set of $50$ numbers, so what does it matter any more where the set "starts"? If I write $\{11,12,13,\ldots,49,50,1,2,3,\ldots,9,10\},$ it means exactly the same thing as $\{1,2,3,\ldots,49,50\}.$ The "starting" point of the sum of a finite set is just as meaningless, since addition is commutative.

This logic does not prove that my interpretation is correct--I believe the wording is ambiguous--but I find the other interpretation a highly unnatural use of the natural English language.

Answer 5

The question says to calculate the sum of 50 natural numbers from 11 , it doesn't mean that to calculate the sum of natural numbers from 11 to 50 . So there is an high school equation to calculate this its is Sn = n/2 (2f+ (n -1)d) S - sum of natural numbers n - no. Of terms ( eg: in this situations we are going to calculate the sum of 50 terms) f - First term ( eg : here 11 is the first term because we are adding the natural numbers starting frm 11) (n - 1) - we have to subtract 1 from the no. Of terms .( eg: here no. Of terms we are going to add is 50 , so we need to subtract 1 from it (50 - 1) and we get 49 )

d - it is the commen difference between terms ( here the first term is 11 and 2nd term will be 12 . So difference between terms is 12- 11=1) Therefore ans; S50 = 50 /2( 2×11+49×1) =25(22+49) = 25 × 71 = 1775