Determine
208/207+(208/207×206/205)+(208/207×206/205×204/203)+..+(208/207×206/205×204/203..4/3*2/1)
208
208
To determine the value of the expression:
208/207 + (208/207 × 206/205) + (208/207 × 206/205 × 204/203) + ... + (208/207 × 206/205 × 204/203 × ... × 4/3 × 2/1)
We can approach this problem by evaluating the terms step by step.
Let's break down the expression:
Term 1: 208/207
Term 2: 208/207 × 206/205
Term 3: 208/207 × 206/205 × 204/203
...
Term n: 208/207 × 206/205 × ... × 4/3 × 2/1 (n terms)
We can notice a pattern here. Each term is formed by multiplying the previous term by a fraction that decreases by one in both the numerator and denominator.
We can rewrite the ith term as:
(208/207) × (206/205) × ... × (4/3) × (2/1)
-----------------------------------------
(207/207) × (205/205) × ... × (3/3) × (1/1)
Simplifying, we get:
(208 × 206 × ... × 4 × 2)
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(207 × 205 × ... × 3 × 1)
This can be simplified further:
(208 × 206 × ... × 4 × 2)
---------------------------
(207 × 205 × ... × 3 × 1)
Now, we can cancel out some terms:
(208 × 206 × ... × 4 × 2)
---------------------------
(207 × 205 × ... × 3 × 1)
= (2 × 2)
----------
(3 × 1)
= 4/3
Therefore, the value of the given expression is 4/3.