Find the term indicated in the expansion. (2x-3)^6; fifth term

Question

Accepted Answer

Hello. Today we're gonna be figuring out what the seventh term of the binomial is. So before we jump into this problem, let's just go ahead and quickly recall what the binomial theorem is. So what the binomial theorem states is that if you have a binomial in the form of A plus B raised to the power of N. This could be rewritten as the some nation as K is equal to zero. All the way to end of the binomial coefficient. And Kay times a race to the power of n minus K times B raised to the power of K. And this is also where the binomial coefficient and K is equal to N. Factorial over N minus K. Factorial times K factorial. Okay, so the instruction state for us to find the seventh term. So before we find the seventh term, let's go ahead and figure out what our summation is gonna be. So looking at the binomial given to us, the binomial is four Y minus seven. Raised to the eighth. Power for y is in the spot of A. So A is going to equal to four, Y. Negative seven is in the spot of B. So B is going to equal to negative seven. And n is just the power that this binomial is raised to which is going to be eight. So N is equal to eight. And rewriting this in the form of a summation is going to give us a summation where K is equal to zero all the way to eight. With the binomial coefficient of 80 times A. Which is four Y Race the power of n which is eight minus k times B which is negative seven race of the power of K. Okay, so keep in mind we're looking for the seventh term of this summation. Now a common mistake is going to be that a lot of people are gonna assume that we want to find us the term when K is equal to seven. But this is not the case because keep in mind this summation starts at zero. So all the values from K equals to zero to K is equal to eight is going to be 01,  aN:aN:000NaN and eight. And the seventh term in this summation is actually going to be six. So we want to find the term when K is equal to six. So when K is equal to six, this summation is going to equal to the expansion of the coefficient which is n factorial or eight factorial over n minus K which is eight minus six factorial times six factorial which is K factorial. And that's going to be multiplied by a which is four Y raised to the power of eight minus K which is eight minus six times negative seven. Raised to the power of K which is six. And now we want to just go ahead and simplify this. Well eight minus six is going to give me two factorial And a factorial could be rewritten as eight times seven times 6 factorial doing this allows me to cancel out a six factorial in both the numerator and denominator and two factorial is the same thing as to now four Y race. The power of eight minus 68 minus six is just going to be two and four Y raised to the power of two is going to be 16 Y squared. And we're multiplying that by -7 to the 6th. Power. Okay, now putting everything together is going to give us eight times 7/2 times 16 Y squared times negative seven raised to the power of six. And multiplying eight times seven times times negative seven to the power of six. And then dividing everything by two is going to give me the value of 52,706, times Y squared. And that is going to be the solution for 1/7 term. And that also means that the answer to this problem is going to be d. So, I hope this video helps you in understanding how to find a specific term of the binomial theorem. And I'll go ahead and see you all in the next video

Find the term indicated in the expansion. (2x-3)^6; fifth term

Key Concepts

Binomial Theorem

Binomial Coefficients

Term Position in Binomial Expansion

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