The sum of all rational terms in the expansion of (2+3)8 is:
Get the complete, step-by-step math solution for: "The sum of all rational terms in the expansion of (2+√(3))^8 is:". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Identify the general term
We use the binomial theorem to find the general term of the expansion (2+3)8. Here, n=8, a=2, and b=3. The general term is given by the formula Tr+1=(rn)an−rbr.
Step 2: Substitute values into the general term
Substituting the values n=8, a=2, and b=3 into the general term formula, we get Tr+1=(r8)(2)8−r(3)r. We can simplify (3)r as 3r/2.
Step 3: Determine conditions for rational terms
For the term Tr+1 to be rational, the power of 3 must be an even integer. This means r must be an even integer, since r can range from 0 to 8. So, possible values for r are 0,2,4,6,8.
Step 4: Calculate rational terms for each 'r'
Now we calculate each rational term by substituting the even values of r into the general term formula. We find T1 (for r=0), T3 (for r=2), T5 (for r=4), T7 (for r=6), and T9 (for r=8).
Step 5: Sum the rational terms
Finally, we sum all the calculated rational terms to find the total sum of all rational terms in the expansion.