What is the value of the 3rd term of the expansion $(3 y-2)^7$?
Answer (2)
Use the binomial theorem to express the general term: ( k n ) a n − k b k .
Identify n = 7 , a = 3 y , b = − 2 , and k = 2 for the 3rd term.
Calculate the binomial coefficient: ( 2 7 ) = 21 .
Compute the 3rd term: 21 \t ( 3 y ) 5 ( − 2 ) 2 = 20412 y 5 , so the answer is 20412 y 5 .
Explanation
Understanding the Problem We are asked to find the 3rd term in the binomial expansion of ( 3 y − 2 ) 7 . Let's recall the binomial theorem, which provides a way to expand expressions of the form ( a + b ) n .
Recalling the Binomial Theorem The general term in the binomial expansion of ( a + b ) n is given by the formula: ( k n ) a n − k b k where ( k n ) is the binomial coefficient, also written as C ( n , k ) or n C k , and k ranges from 0 to n . In our case, we have a = 3 y , b = − 2 , and n = 7 .
Finding the Correct Value of k Since we want to find the 3rd term, we need to determine the value of k . Remember that the first term corresponds to k = 0 , the second term to k = 1 , and so on. Therefore, the 3rd term corresponds to k = 2 .
Substituting Values into the Formula Now, we substitute the values n = 7 , k = 2 , a = 3 y , and b = − 2 into the general term formula: ( 2 7 ) ( 3 y ) 7 − 2 ( − 2 ) 2 Let's calculate each part separately.
Calculating the Binomial Coefficient First, we calculate the binomial coefficient ( 2 7 ) :
( 2 7 ) = 2 ! ( 7 − 2 )! 7 ! = 2 ! 5 ! 7 ! = 2 × 1 7 × 6 = 21 So, ( 2 7 ) = 21 .
Calculating (3y)^5 Next, we calculate ( 3 y ) 7 − 2 = ( 3 y ) 5 :
( 3 y ) 5 = 3 5 y 5 = 243 y 5 So, ( 3 y ) 5 = 243 y 5 .
Calculating (-2)^2 Then, we calculate ( − 2 ) 2 :
( − 2 ) 2 = 4 So, ( − 2 ) 2 = 4 .
Multiplying All Terms Together Finally, we multiply all the terms together: 21 × 243 y 5 × 4 = 20412 y 5 Therefore, the 3rd term of the expansion ( 3 y − 2 ) 7 is 20412 y 5 .
Final Answer The value of the 3rd term of the expansion ( 3 y − 2 ) 7 is 20412 y 5 .
Examples
Binomial expansions are used in various fields such as probability, statistics, and physics. For instance, in probability, when calculating the likelihood of a certain number of successes in a series of independent trials, the binomial theorem can be applied. In physics, it appears in approximations and series expansions to simplify complex equations. Understanding binomial expansions helps in modeling and solving problems in these areas.
The 3rd term of the expansion ( 3 y − 2 ) 7 is found using the binomial theorem, resulting in a value of 20412 y 5 . This term is calculated by determining the appropriate binomial coefficient and substituting the values for a and b . Hence, the final answer is 20412 y 5 .
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