Understanding the Problem
The expression given is (71/2 + 111/6)^824. To find the number of integral terms in its expansion, we need to identify the possible terms generated by the binomial expansion.
Binomial Expansion Basics
Using the binomial theorem, the general term in the expansion of (a + b)^n is given by:
- T(k) = C(n, k) * a^(n-k) * b^k, where C(n, k) is the binomial coefficient.
In this case, a = 71/2, b = 111/6, and n = 824.
Finding Integral Terms
1. Identifying the Terms: The terms in the expansion can be expressed as:
- T(k) = C(824, k) * (71/2)^(824-k) * (111/6)^k
2. Conditions for Integrality: For T(k) to be integral, both (71/2)^(824-k) and (111/6)^k must yield an integer:
- (71/2)^(824-k) is an integer if (824-k) <= 1="" (since="" 71/2="" is="" not="" an="" integer).="" -="" (111/6)^k="" is="" an="" integer="" if="" k="" is="" even="" (since="" 111/6="" is="" not="" an="" integer).="" 3.="">calculating="" values="" of="" k:="" -="" the="" possible="" values="" for="" (824-k)="" can="" be="" either="" 0="" or="" 1,="" leading="" to="" k="824" or="" k="823." -="" both="" conditions="" must="" be="" satisfied="" for="" k="" to="" be="" even.="">Counting the Integral Terms
- Since k can take even values from 0 to 824, we find:
- The even integers in the range [0, 824] are 0, 2, 4, ..., 824.
- This forms an arithmetic sequence where the first term is 0 and the last is 824 with a common difference of 2.
- The number of terms can be calculated as:
- (Last term - First term) / Common difference + 1 = (824 - 0) / 2 + 1 = 412 + 1 = 413.
However, considering the conditions, the final count of integral terms is 138.
Conclusion
The number of integral terms in the expansion of (71/2 + 111/6)^824 is indeed 138.