Let f(x)=22x+1+2x+4+322x+2+16. Then the value of 8(f(151)+f(152)+…+f(1559)) is equal to
Get the complete, step-by-step math solution for: "Let f(x) = {2^{x+2} + 16}{2^{2x+1} + 2^{x+4} + 32}. Then the value of 8f(1)/(15) + f(2)/(15) + + f(59)/(15) is equal to". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Simplify the function f(x)
First, we simplify the given function f(x). We can rewrite the terms in the numerator and denominator using the exponent rule am+n=am⋅an. This allows us to express all terms in terms of 2x.
Step 2: Factor out common terms
Next, we factor out common terms from the numerator and denominator. In the numerator, we factor out 4. In the denominator, we factor out 2 and recognize that the remaining quadratic expression is a perfect square trinomial, (2x+4)2.
Step 3: Further simplify f(x)
We can cancel out one factor of (2x+4) from the numerator and denominator, simplifying the function to f(x)=2x+42. This is the most simplified form of the function.
Step 4: Find the relationship between f(x) and f(4-x)
Now, let's evaluate f(4-x) to see if there's a useful relationship. We substitute (4−x) into the simplified function. After algebraic manipulation, we find that f(x)+f(4−x)=2x+42+8+2⋅2x2x=2x+42+2(4+2x)2x=2(2x+4)4+2x=21.
Step 5: Pair the terms in the sum
The sum is S=f(151)+f(152)+…+f(1559). We notice that the terms can be paired such that the sum of the arguments is 4. For example, 151+1559=1560=4. There are 29 such pairs, and one middle term f(1530)=f(2).
Step 6: Calculate the sum
Each pair sums to 21. There are 29 such pairs. The middle term is f(2)=22+42=4+42=82=41. So the total sum is 29⋅21+41=458+41=459.
Step 7: Calculate the final value
Finally, we need to multiply the sum S by 8 to get the required value.