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Suppose you have a system of 50 linear equations. It's tedious and impractical to solve them by hand. Polynomial interpolation is your better alternative.

\(f(x)\) is a \(5^\text{th}\) degree polynomial such that \(f(1)=2,\) \(f(2)=3,\) \(f(3)=4,\) \(f(4)=5,\) \(f(5)=6,\) and \(f(8)=7.\)

If the value of \(f(9)\) can be expressed as \(\dfrac{a}{b}\) for coprime positive integers \(a\) and \(b\), find the value of \(a+b\).

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\(f(x)\) is a polynomial with integer coefficients. We have,

\(f(1)=1\)

\(f(2)=4\)

\(f(3)=9\)

\(f(4)=16\)

\(f(5)=25\)

\(f(6)=36\)

\(f(7)=49\)

\(f(8)=64\)

\(f(9)=81\)

\(f(10)=100\)

Determine \(f(11)\).

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Roopesh went to his friends house. There he ate a lot of Ice-cream. First he started with Vanilla then Chocolate then Vanilla then Butterscotch then Vanilla. He is a mathematician and made a monic polynomial of degree 5 which gave him values of the first letter of the Ice Cream.

This is the way his polynomial proceeded :-

\(p(1) = 22\)

\(p(2) = 3\)

\(p(3) = 22\)

\(p(4) = 2\)

\(p(5) = 22\)

If the value of \(p(6) = \overline{abc}\). The first letter of the ice cream he eats would be \(a+b+c\). Which ice cream will be eat next?

**Assumption:** \(\overline{abc}\) represents a 3 digit number.

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\[ \large f(1) = 4, f(2) = 9, f(3) = 20, f(4) = 44, f(5) = 88 \]

Let \(f(x)\) be a 5th degree monic polynomial such that it satisfies the equations above. Find the value of \(f(6)\).

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