Show that the number 2(1+2+3+4+5+...+2021+2022)+2023 is a perfect square.
#Help with a olympiad question.
jagged fog
tulip bone
Show that the number 2(1+2+3+4+5+...+2021+2022)+2023 is a perfect square.
do you know AP
arithmetic progression
sum of AP has a formula
$$\frac{n}{2}(a+l)$$
mighty plankBOT
AKG
tulip bone
n -> no. of terms
a -> first term
l -> last term
we'll calculate this first with the above formula
,w 2022/2 * (1+2022)
mighty plankBOT
tulip bone
,calc sqrt(2*2045253 + 2023)
mighty plankBOT
Result:
2023
hence perfect square
jagged fog
**AKG**
Oh. So this formula works for all of the exercises that are needed to find the sum of a large number of numbers in the parenthesis?
Like (1+3+5+7+9+...+366)?
tulip bone
Like (1+3+5+7+9+...+366)?
not for this one
there's another
$$\frac{n}{2}(2a+(n-1)d)$$
mighty plankBOT
AKG
tulip bone
n -> of terms
a -> first term
d -> common difference between 2 terms
jagged fog
Is there another formula for a common difference between 3 terms or 4?
Or do I just substitute d with the common difference in each case
tulip bone
$$T_n = a + (n-1)d$$
mighty plankBOT
AKG
jagged fog
Oh okay
jagged fog
Tn -> nth term
a -> first term
d -> common difference
What do you mean by "nth term"?
tulip bone
What do you mean by "nth term"?
for exmaple we have an AP: 2, 6, 10, 14, ....
if we talk about T3 it means 3rd term which is 10 here
jagged fog
Oh, okay
jagged fog
,calc sqrt(2*2045253 + 2023)
But how am I supposed to know that that number is a square root?
tulip bone
But how am I supposed to know that that number is a square root?
every real number has a square root, we were talking about perfect square
let's have a look at perfect square's definition
jagged fog
Oh, yeah, that's what I wanted to say, my bad.
tulip bone
,w perfect square
mighty plankBOT
tulip bone
for example: 25 is a perfect square because it's square root is a whole number ie 5
but 2 is not a perfect square it's square root is 1.4142135624... which is not a whole number
jagged fog
I get it, but this number is too large for me to know. 25 and other numbers like 36, 49, 64,81, 100,121 are learned
tulip bone
I get it, but this number is too large for me to know. 25 and other numbers like...
we use prime factorization method to calculate square roots
example:
324's square root will be multiplication of the common pairs which is 2 and 3s above
2*3*3 => 18
now there's a chance that we won't get common pair for example 8:
we got a pair but (purple line) but still one number is left alone (green circle), so 8 is not a perfect square