if the coef X¹⁰ .(ax² +1/bx)¹¹ Is equalX⁻¹⁰ (ax -1/bx²)¹¹Find the relation of a,b real numbers.

In this video, we will be exploring the coefficients of two different expansions and finding the relation between two real numbers, a and b. Specifically, we will be looking at the coefficient of x^10 in the expansion of (ax^2 +1/bx)^11 and comparing it to the coefficient of x^(-10) in the expansion of (ax -1/bx^2)^11. Through our exploration, we will discover the relationship between a and b that satisfies this condition.
Background Information:
Before we delve into the mathematics, let us first establish some basic knowledge on binomial expansions. The binomial theorem states that for any positive integer n and any real numbers a and b, the expansion of (a+b)^n is given by the following formula:
(a+b)^n = nC0(a^n)(b^0) + nC1(a^(n-1))(b^1) + nC2(a^(n-2))(b^2) + ... + nCr(a^(n-r))(b^r) + ... + nCn(a^0)(b^n)
where nCr denotes the number of ways of choosing r objects from a set of n objects. This formula provides a way to expand expressions that are raised to a power.
Problem Statement:
Now that we have established the binomial theorem, let us look at the problem at hand. We are given two expressions: (ax^2 +1/bx)^11 and (ax -1/bx^2)^11. We are asked to find the relation between a and b that satisfies the condition where the coefficient of x^10 in the first expression is equal to the coefficient of x^(-10) in the second expression.
Solution:
To solve this problem, we will use the binomial theorem to expand both expressions and compare the coefficients of x^10 and x^(-10). Let us begin by expanding (ax^2 +1/bx)^11.
(ax^2 +1/bx)^11 = 11C0(a^11)(x^22/b^0) + 11C1(a^10)(1/b^1)(x^20) + 11C2(a^9)(1/b^2)(x^18) + ... + 11C10(a^1)(1/b^10)(x^2) + 11C11(1/b^11)(x^0)
Notice that the powers of x decrease by 2 as we move from term to term. This is because we have (ax^2) and (1/bx) in the expression, and when multiplied together, we get x^2/b. Therefore, when we expand the expression, the powers of x will decrease by 2.
Now, let us expand (ax -1/bx^2)^11.
(ax -1/bx^2)^11 = 11C0(a^11)(x^11) + 11C1(a^10)(-1/b^1)(x^9) + 11C2(a^9)(1/b^2)(x^7) + ... + 11C10(a^1)(-1/b^10)(x^(-9)) + 11C11(-1/b^11)(x^(-22))
Notice that the powers of x increase by 2 as we move from term to term. This is because we have (ax) and (-1/bx^2) in the expression, and when multiplied together, we get -x/b. Therefore, when we expand the expression, the powers of x will increase by 2.
Now that we have both expansions, we can compare the coefficients of x^10 and x^(-10).
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