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 in the expansion of ( 2 x + y ) 9 .
Identify the 7th term in the expansion, which corresponds to k = 6 in the binomial coefficient ( k 9 ) .
Calculate the binomial coefficient ( 6 9 ) = 84 .
Determine the 7th term to be 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 . We can use the binomial theorem to find this term.
Recalling the Binomial Theorem The binomial theorem states that ( a + b ) n = ∑ k = 0 n ( k n ) a n − k b k . The k -th term in the expansion of ( a + b ) n is given by ( k − 1 n ) a n − ( k − 1 ) b k − 1 .
Applying the Binomial Theorem to our problem In our case, a = 2 x , b = y , and n = 9 . We want to find the 7th term, so k = 7 . Thus, the 7th term is given by ( 7 − 1 9 ) ( 2 x ) 9 − ( 7 − 1 ) y 7 − 1 = ( 6 9 ) ( 2 x ) 9 − 6 y 6 = ( 6 9 ) ( 2 x ) 3 y 6 .
Calculating the binomial coefficient Now we need to calculate ( 6 9 ) . Using the formula for binomial coefficients, we have ( 6 9 ) = 6 ! 3 ! 9 ! = 3 × 2 × 1 9 × 8 × 7 = 3 × 4 × 7 = 84.
Calculating (2x)^3 Next, we calculate ( 2 x ) 3 = 2 3 x 3 = 8 x 3 .
Finding the 7th term Therefore, the 7th term is 84 ( 8 x 3 ) y 6 = 672 x 3 y 6 .
Examples
The binomial theorem is not just an abstract mathematical concept; it has practical applications in various fields. For instance, in probability, it helps calculate the likelihood of different outcomes in a series of independent trials, like coin flips or dice rolls. In finance, it can be used to model investment growth and predict future values based on certain probabilities. Moreover, in physics, the binomial theorem appears in approximations and expansions used to simplify complex equations, making it a versatile tool for problem-solving across disciplines. For example, when calculating probabilities, consider a scenario where you want to know the chance of getting exactly 3 heads when flipping a coin 5 times. The binomial theorem provides a straightforward way to compute this probability: P ( X = k ) = ( k n ) p k ( 1 − p ) n − k , where n is the number of trials, k is the number of successes, and p is the probability of success on a single trial.
The 7th term in the expansion of ( 2 x + y ) 9 is calculated using the binomial theorem. After applying the formula and calculating the necessary components, we find that the 7th term is 672 x 3 y 6 . The final answer is 672 x 3 y 6 .
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