Final answer:
The term containing b¹⁰ in the expansion of (a+b²)¹⁸ is found using the binomial theorem. Since b² must be raised to the power of 5 to get b¹⁰, the coefficient is ¹⁸C⁵, which equals 8568. Thus, the term is 8568a¹³b¹⁰.
Step-by-step explanation:
The student is asking for help with finding the term containing b¹⁰ in the expansion of the binomial (a+b²)¹⁸. To solve this, we use the binomial theorem, which states that when a binomial expression is raised to an nth power, the expansion will be the sum of n+1 terms, where each term is of the form:
π⁼ = ⁿCₘ * aⁿ⁹ₘ * b²ₘ
(Where ⁿCₘ is the binomial coefficient, given by ⁿ! / (ₘ! * (ⁿ-ₘ)!))
In the expansion of (a+b²)¹⁸, to find the term containing b¹⁰, we need to determine the appropriate k for which b¹⁰ appears. Since the binomial involves b², we need b² raised to the 5th power to get b¹⁰ (because 2*5 = 10). Therefore, we are looking for the term where k = 5, which is the 6th term in the sequence (since we start counting from k = 0).
The coefficient of this term can be calculated using the binomial coefficient formula: ¹⁸C⁵ = 18! / (5!*(18-5)!) = 8568. So the term we're looking for is 8568a¹³b¹⁰.