What is the value of the 7th term in [tex]$(2 x+y)^9$[/tex]?
Answer (2)
Use the binomial theorem to express the general term: T k + 1 = ( k n ) a n − k b k .
Identify n = 9 , a = 2 x , b = y , and k = 6 for the 7th term.
Calculate the binomial coefficient: ( 6 9 ) = 84 .
Simplify the 7th term: T 7 = 672 x 3 y 6 , so the final answer is 672 x 3 y 6 .
Explanation
Understanding the Problem We are asked to find the 7th term in the expansion of ( 2 x + y ) 9 . The binomial theorem provides a way to expand expressions of the form ( a + b ) n . The general term in the binomial expansion of ( a + b ) n is given by T k + 1 = ( k n ) a n − k b k , where k starts from 0. In our case, a = 2 x , b = y , and n = 9 . We want to find the 7th term, which corresponds to k + 1 = 7 , so k = 6 .
Applying the Binomial Theorem Using the binomial theorem, the 7th term in the expansion of ( 2 x + y ) 9 is given by: T 7 = ( 6 9 ) ( 2 x ) 9 − 6 ( y ) 6 We need to calculate the binomial coefficient ( 6 9 ) and simplify the expression.
Calculating the Binomial Coefficient First, let's calculate the binomial coefficient ( 6 9 ) .
( 6 9 ) = 6 ! 3 ! 9 ! = 3 × 2 × 1 9 × 8 × 7 = 6 9 × 8 × 7 = 3 × 4 × 7 = 84 So, ( 6 9 ) = 84 .
Simplifying the Expression Next, let's simplify ( 2 x ) 9 − 6 = ( 2 x ) 3 .
( 2 x ) 3 = 2 3 x 3 = 8 x 3
Finding the 7th Term Now, substitute the values back into the expression for T 7 :
T 7 = ( 6 9 ) ( 2 x ) 3 y 6 = 84 × 8 x 3 y 6 = 672 x 3 y 6 Therefore, the 7th term in the expansion of ( 2 x + y ) 9 is 672 x 3 y 6 .
Final Answer The 7th term in the expansion of ( 2 x + y ) 9 is 672 x 3 y 6 .
Examples
Binomial expansions are used in probability calculations, such as determining the likelihood of specific outcomes in a series of independent trials. For example, if you flip a coin 9 times, the binomial expansion can help you calculate the probability of getting exactly 6 heads. The coefficients in the expansion relate to the number of ways each outcome can occur, and each term represents a different probability scenario.
The 7th term in the expansion of ( 2 x + y ) 9 is 672 x 3 y 6 . This is found using the binomial theorem, calculating the binomial coefficient and simplifications. The binomial theorem allows us to express the expansion of polynomials systematically.
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