What is the fifth term in the binomial expansion of $(x+5)^8$?
Answer (1)
The general term in the binomial expansion of ( a + b ) n is ( k n ) a n − k b k .
Substitute a = x , b = 5 , n = 8 , and k = 4 to find the fifth term: ( 4 8 ) x 8 − 4 5 4 .
Calculate the binomial coefficient: ( 4 8 ) = 70 .
Calculate 5 4 = 625 , and multiply by the binomial coefficient to get the fifth term: 70 × 625 x 4 = 43 , 750 x 4 .
Explanation
Understanding the Binomial Theorem We are asked to find the fifth term in the binomial expansion of ( x + 5 ) 8 . Let's recall the binomial theorem, which tells us how to expand expressions of the form ( a + b ) n . The general term in the binomial expansion of ( a + b ) n is given by ( k n ) a n − k b k , where ( k n ) = k ! ( n − k )! n ! is a binomial coefficient.
Identifying the Correct Term In our case, we have a = x , b = 5 , and n = 8 . We want to find the fifth term in the expansion. Since the expansion starts with k = 0 , the fifth term corresponds to k = 4 .
Substituting Values Now, we substitute a = x , b = 5 , n = 8 , and k = 4 into the general term formula: ( 4 8 ) x 8 − 4 5 4 .
Calculating the Binomial Coefficient Let's calculate the binomial coefficient ( 4 8 ) . We have ( 4 8 ) = 4 ! 4 ! 8 ! = ( 4 × 3 × 2 × 1 ) ( 4 × 3 × 2 × 1 ) 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 4 × 3 × 2 × 1 8 × 7 × 6 × 5 = 70.
Calculating the Power of 5 Next, we calculate 5 4 = 5 × 5 × 5 × 5 = 625 .
Finding the Fifth Term Now, we multiply the binomial coefficient and the power of 5: 70 × 625 = 43750 . Therefore, the fifth term is 43750 x 8 − 4 = 43750 x 4 .
Final Answer The fifth term in the binomial expansion of ( x + 5 ) 8 is 43 , 750 x 4 .
Examples
Binomial expansion is used in probability calculations, such as determining the likelihood of different outcomes in a series of independent trials. For example, if you flip a coin 8 times, the binomial expansion can help you calculate the probability of getting exactly 4 heads. It also has applications in physics, statistics, computer science, and finance, allowing us to model and understand complex systems and predict future outcomes based on probabilities and combinations.
