(a+b+cc)^(2017)+(a-b-c)^(2017)
When simplified how many terms would that be?
can we try this? i saw it in IMO past question I couldn't do it
Sir obleck I search for that book you said ,but no luck for me ....I only have 2014 past question but it hasn't been helpful so far...I need to understand the ideal and fast thinking to question like this.....
(a+b+c)^1 = 2a
(a+b+c)^3 + (a-b-c)^3 = 2a(a^2 + 3(b+c)^2)
(a+b+c)^5 + (a-b-c)^5 = 2a(a^4 + 10a^2(b+c)^2 + 5(b+c)^4)
So, for k=n, there are
1: 1
3: 1+3
5: 1+3+5
these are the odd squares.
The answer must be 2017^2
Just the right person I was looking for thanks man
To simplify the expression (a+b+cc)^(2017) + (a-b-c)^(2017), we need to expand each term and consider the resulting terms of the binomial expansion.
Let's start by expanding the first term: (a+b+cc)^(2017).
Using the binomial expansion formula, we get:
(a+b+cc)^(2017) = C(2017,0)a^(2017)(b+cc)^0 + C(2017,1)a^(2016)(b+cc)^1 + C(2017,2)a^(2015)(b+cc)^2 + ... + C(2017,2017)(b+cc)^2017.
Similarly, expanding the second term (a-b-c)^(2017), we get:
(a-b-c)^(2017) = C(2017, 0)a^(2017)(-b-c)^0 + C(2017, 1)a^(2016)(-b-c)^1 + C(2017, 2)a^(2015)(-b-c)^2 + ... + C(2017, 2017)(-b-c)^2017.
The number of terms in the simplified expression is determined by the number of unique combinations of the exponents of a, b, and c. To find this, let's analyze the possible combinations term by term.
For the first term (a+b+cc)^(2017):
When expanding (b+cc)^0, we have one term.
When expanding (b+cc)^1, we have two terms (one term from b and one term from cc).
When expanding (b+cc)^2, we have three terms (b^2, 2bcc, and c^2).
In general, for (b+cc)^n, we would have n+1 terms.
So, the number of terms in (a+b+cc)^(2017) would be 2017+1 = 2018.
Similarly, for the second term (a-b-c)^(2017):
When expanding (-b-c)^0, we have one term.
When expanding (-b-c)^1, we have two terms (-b and -c).
When expanding (-b-c)^2, we have three terms (b^2, 2bc, and c^2), after considering the negative signs.
In general, for (-b-c)^n, we would have n+1 terms.
Thus, the number of terms in (a-b-c)^(2017) would also be 2017+1 = 2018.
Now, when we add the two expanded expressions together, we need to consider the possible combinations of terms between these two expanded expressions.
Since both expressions have the same number of terms (2018 terms each), and each term is a unique combination of exponents of a, b, and c, adding the two expressions will still result in 2018 terms.
Therefore, when simplified, the expression (a+b+cc)^(2017) + (a-b-c)^(2017) will have a total of 2018 terms.