Algebra (Olympiad)mediumFree

Evaluate (1^4+2007^4+2008^4)/(1^2+2007^2+2008^2) | IOQM

JEE Maths question with a full step-by-step solution.

Question
Find the value of
14+20074+2008412+20072+20082,\quad \frac{1^4 + 2007^4 + 2008^4}{1^2 + 2007^2 + 2008^2}, \qquad\qquad
Solution
Answer: 4030057
For all real a,ba, b,
a4+b4+(a+b)4=12[a2+b2+(a+b)2]2a^4 + b^4 + (a+b)^4 = \frac{1}{2}\big[a^2 + b^2 + (a+b)^2\big]^2
Since 2008=1+20072008 = 1 + 2007, take a=1a = 1, b=2007b = 2007, a+b=2008a + b = 2008.
14+20074+2008412+20072+20082=12[12+20072+20082]212+20072+20082=12(1+20072+20082)\frac{1^4 + 2007^4 + 2008^4}{1^2 + 2007^2 + 2008^2} = \frac{\tfrac{1}{2}\big[1^2 + 2007^2 + 2008^2\big]^2}{1^2 + 2007^2 + 2008^2} = \frac{1}{2}\big(1 + 2007^2 + 2008^2\big)
 With 20072=40280492007^2 = 4028049 and 20082=40320642008^2 = 4032064,
=12(1+4028049+4032064)=12(8060114)=4030057= \frac{1}{2}(1 + 4028049 + 4032064) = \frac{1}{2}(8060114) = 4030057
Still stuck on this question?Ask your doubt on WhatsApp
Similar questions
Algebra (Olympiad) · hard
Let {an}\{a_n\} be a sequence of positive integers such that a1=a2=2a_1=a_2=2. For n≥1n\ge1, an+2an−an+12=an+1an.a_{n+2}a_n-a_{n+1}^2=a_{n+1}a_n. Determine the value of a1a2a1032 a7\dfrac{a_1a_2a_{10}}{32\,a_7}.
Algebra (Olympiad) · hard
Consider the polynomial (x2+1)(x2+4)(x2−2x+2)(x2+2x+2).(x^2+1)(x^2+4)(x^2-2x+2)(x^2+2x+2). Let f(x)=(x4+2x2+2x)2+(P(x))2f(x)=\big(x^4+2x^2+2x\big)^2+\big(P(x)\big)^2, where P(x)P(x) is a cubic polynomial. Find the value of P(1) P(2)P(1)\,P(2)
Algebra (Olympiad) · hard
Let f(n)=12n3−5n2−251n+3896n2−37n+45.f(n) = \frac{12n^3 - 5n^2 - 251n + 389}{6n^2 - 37n + 45}. There is a unique positive integer nn for which f(n)f(n) is an integer. Denote this unique value of nn by pp. Find the value of p+f(p)p + f(p).
Algebra (Olympiad) · hard
Let x=1+112+122+1+122+132+1+132+142+⋯+1+1992+11002.x=\sqrt{1+\frac{1}{1^2}+\frac{1}{2^2}}+\sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+\sqrt{1+\frac{1}{3^2}+\frac{1}{4^2}}+\cdots+\sqrt{1+\frac{1}{99^2}+\frac{1}{100^2}}. If [x][x] denotes the greatest integer not exceeding xx, then find the value of 100 (x−[x])100\,(x-[x])

Solve more, learn faster

Sign up free to solve more JEE Maths questions and explore doMath — timed drills, mastery sprints, bookmarks, and chapter-wise progress tracking.

Sign up freeExplore doMath