If the sum of the first 10 terms of the series
1+4⋅144⋅1+1+4⋅244⋅2+1+4⋅344⋅3+… is nm, where gcd(m, n) = 1, then m + n is equal to:
Get the complete, step-by-step math solution for: "If the sum of the first 10 terms of the series (4 · 1)/(1 + 4 · 1^4) + (4 · 2)/(1 + 4 · 2^4) + (4 · 3)/(1 + 4 · 3^4) + is (m)/(n), where gcd(m, n) = 1...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Identify the general term
First, we need to identify the general term of the given series. By observing the pattern in the numerators and denominators, we can express the k -th term as Tk=1+4k44k. This formula accurately represents each term in the series.
Step 2: Rewrite the denominator using Sophie Germain Identity
The denominator 1+4k4 can be factored using the Sophie Germain Identity, which states a4+4b4=(a2+2b2+2ab)(a2+2b2−2ab). Here, we can let a=1 and b=k. This factorization is crucial for simplifying the term.
Step 3: Decompose the general term using partial fractions
Now, we can decompose the general term Tk into a difference of two fractions. Notice that the numerator 4k is the difference between the two factors in the denominator: (2k2+2k+1)−(2k2−2k+1)=4k. This allows us to express Tk as a telescoping sum.
Step 4: Define a new function for simplification
Let's define f(k)=2k2−2k+1. Then, we can observe that f(k+1)=2(k+1)2−2(k+1)+1=2(k2+2k+1)−2k−2+1=2k2+4k+2−2k−2+1=2k2+2k+1. This means our general term can be written as Tk=f(k)1−f(k+1)1.
Step 5: Calculate the sum of the first 10 terms
The sum of the first 10 terms is a telescoping series. When we expand the sum, intermediate terms cancel out. We are left with the first term of the first fraction and the last term of the second fraction. We calculate f(1)=2(1)2−2(1)+1=1 and f(11)=2(11)2−2(11)+1=2(121)−22+1=242−22+1=221.
Step 6: Compute the final sum and m + n
Substituting the values of f(1) and f(11) into the sum, we get S10=1−2211=221221−1=221220. Here, m=220 and n=221. Since gcd(220,221)=1, we can find m+n=220+221=441.