D {(1,-4),(-2,-4),(8,3),(10,3)}
Let 
be the first term in the series and 
the common ratio. Then the infinite series converges to
Removing the first three terms from the left side effectively multiplies the right side by 27, so
By elimination,
Solve for . We can eliminate so that
Answer:8.385
Step-by-step explanation:
Step 1: multiply both sides by 30 (this sounds crazy, but that's 5 and 6's least common multiple) 1/6(j-1)+4=1/5(3j+1) -> 5(j-1)+120=6(3j+1)
Step 2: Distribute 5(j-1)+120=6(3j+1) -> 5j-5+120=18j+6
Step 3: Combine like terms 5j-5+120=18j+6 -> 5j+115=18j+6
Step 4: Isolate the variable 5j+115=18j+6 -> 109=13j
Step 5: Divide to further isolate 109/13=j= 8.385
f(a) = 3a² + 8a + 5
substitute x = a into f(x)
f(a) = 3a² + 8a + 5
The Pyth. Thm. tells us the following: (hypo)^2 = (leg 1)^2 + (leg 2)^2.
Here the hypo is 75 miles. One train traveled x miles and the other x+10 miles.
applying the Pyth. Thm.,
x^2 + (x+10)^2 = 75^2
which becomes x^2 + x^2 + 20x + 400 = 5625, or
2x^2 + 20x - 5225 = 0.
Choose a method easy for you that will lead to a solution (x-value).
Using "completing the square," I obtained x= 47.8 miles and x = 57.8 miles. I
It's very important to check one's work. Suppose x = 47.8 miles. Then x+10 = 57.8 miles.
47.8^2 + 57.8^2 = 75^ must be true. Is it?
2284.84 + 3340.84 5625
5625.6 is approx equal to 5625. Close enough.
The first train traveled 47.8 miles and the second 57.8 miles.
