\[\LARGE a^n+b^n+c^n=(a+b+c)(a^{n-1}+b^{n-1}+c^{n-1})-(ab+bc+ac)(a^{n-2}+b^{n-2}+c^{n-2})+abc(a^{n-3}+b^{n-3}+c^{n-3})\]

\[\huge a^n+b^n=(a+b)(a^{n-1}+b^{n-1})-ab(a^{n-2}+b^{n-2})\]

\[\huge a^n+\frac{1}{a^n}=(a+\frac{1}{a})(a^{n-1}+\frac{1}{a^{n-1}})-(a^{n-2}+\frac{1}{a^{n-2}})\]

For exercises: Try this problem

Try this very nice set by Aditya.

And this

## Comments

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TopNewestCan we use newton sums for more then three unknowns also . . I mean can we do this:- \[(a^4+b^4+c^4+d^4)=(a+b+c+d)(a^3+b^3+c^3+d^3)-(ab+bc+bd+ac+cd+ad)(a^2+b^2+c^2+c^2)+abcd(a+b+c+d)\] – Aman Sharma · 2 years, 2 months ago

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– Sanjeet Raria · 2 years, 2 months ago

\[(a^n+b^n+c^n+d^n)=(a+b+c+d)(a^{n-1}+b^{n-1}+c^{n-1}+d^{n-1})-(ab+ac+ad+bc+bd+cd) (a^{n-2}+b^{n-2}+c^{n-2}+d^{n-2})+(abc+acd+bcd+abd) (a^{n-3}+b^{n-3}+c^{n-3}+d^{n-3})-abcd (a^{n-4}+b^{n-4}+c^{n-4}+d^{n-4}) \]Log in to reply

(I am realy sorry to disturb you..... :-( i am very new with maths and want learn it in more depth so please help me) – Aman Sharma · 2 years, 2 months ago

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– Sanjeet Raria · 2 years, 2 months ago

Yes you can generalize this pattern.Log in to reply

– Avn Bha · 2 years, 1 month ago

can you tell how did you derive this patternLog in to reply

Feel free to ask or share anything regarding math. It never bores me. – Sanjeet Raria · 2 years, 2 months ago

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this website i found a different explaination about newton sums please check it out – Aman Sharma · 2 years, 2 months ago

InLog in to reply

– Sanjeet Raria · 2 years, 2 months ago

Yes.. That's why I've named it "in easy notation"Log in to reply

– Aman Sharma · 2 years, 2 months ago

:-) ok thank you so much for helping me going to try out these identities in the questions you attatched with your note ......it is a very helpfull note realyLog in to reply

– Sanjeet Raria · 2 years, 2 months ago

You've to count every case, including the third symmetric sum.Log in to reply

– Dinesh Chavan · 2 years, 2 months ago

Remember it is cyclic. U missed abc+Bca.....Log in to reply

@Sanjeet Raria please help – Aman Sharma · 2 years, 2 months ago

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Can you please give an example of how to use these identities....i am struggling with algebra...... and sorry to disturb you – Aman Sharma · 2 years, 3 months ago

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@Aman Sharma @Pranjali Bhargava – Sanjeet Raria · 2 years, 3 months ago

I attached many questions which uses these identities. Just see my note again. Sorry i could not provide the questions regarding the second identity because i didn't come across any. But one can find questions regarding this in Quadratic equations chapter. Questions are asked to find the symmetric expression of roots etc.Log in to reply

But how can we solve the questions related to this using exponential function's graph. Like if we have 3^x + 4^x + 5^x = 6^x (this is a ques. of A. Dasgupta), then divide on both sides by 6^x...without using the identity – Pranjali Bhargava · 2 years, 3 months ago

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– Sanjeet Raria · 2 years, 3 months ago

Yeah.. There's only graphical approach to such questions that you mentioned, applying algebra would be very lengthy & less reliable.Log in to reply