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What is the value of f(2)/f(1) + f(3)/f(2) + f(4)/f(3) + …

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What is the value of f(2)/f(1) + f(3)/f(2) + f(4)/f(3) + …

by Max@Math Revolution » Thu Oct 03, 2019 4:56 am
[GMAT math practice question]

What is the value of f(2)/f(1) + f(3)/f(2) + f(4)/f(3) + ... +f(2006)/f(2005)?

1) f(1)=1
2) f(a+b)=f(a)f(b)
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by deloitte247 » Sat Oct 05, 2019 9:19 pm
$$\frac{f\left(2\right)}{f\left(1\right)}=>\ This\ \exp ression\ looks\ like\ the\ common\ ratio\ of\ a\ geometric\ sequence.$$
$$If\ first\ term=\frac{f\left(2\right)}{f\left(1\right)};$$
$$Second\ term=\frac{f\left(3\right)}{f\left(2\right)};$$
$$nth\ term=\frac{f\left(n+1\right)}{f\left(n\right)};$$
Then this could be an arithmetic sequence as well. We need to find the value of at least 2 unknown functions to decipher if we will finding the sum of AP or GP.
Statement 1=> f(1)=1
$$f\left(x\right)=x\ or\ f\left(x\right)=x^0\ or\ f\left(x\right)=\left(x+1\right)-1$$
No expression is given to validate this statement if we define any expression that's true for x=1, we might get varying results when x=2. Hence, statement 1 is NOT SUFFICIENT.

Statement 2=> f(g+b) = f(g) * f(b)
i.e f(3) = f(2+1) = f(2) * f(1)
and f(2) = f(1+1) = f(1) * f(1).
This statement tells us that the sum of all the arguments in a function can be expressed as product of a function of its own. But to solve the question with this statement, we need to know the exact value of f(1). And since there is no expression to solve for f(1), statement 2 is NOT SUFFICIENT.

Combining both statements together/u]
S1: f(1)=1
S2: f(g+b) = f(g) * f(b)
$$Therefore,\ f\left(2\right)=f\left(1+1\right)=f\left(1\right)\cdot f\left(1\right)=1\cdot1=1$$ $$f\left(3\right)=f\left(2+1\right)=f\left(2\right)\cdot f\left(1\right)=1\cdot1=1$$
$$f\left(4\right)=f\left(2+2\right)=f\left(2\right)\cdot f\left(2\right)=1\cdot1=1$$
$$f\left(n\right)=1$$
$$So\ therefore,\ f\left(1\right),\ f\left(2\right),\ f\left(3\right),...,f\left(2006\right)\ all\ have\ the\ same\ values\ which\ is\ 1.$$
$$The\ value\ of\ \frac{f\left(2\right)}{f\left(1\right)}=\frac{1}{1}=1$$
$$\frac{f\left(3\right)}{f\left(2\right)}=\frac{1}{1}=1$$
$$With\ careful\ observation\ from\ \frac{f\left(2\right)}{f\left(1\right)}to\frac{f\left(5\right)}{f\left(4\right)},$$
$$\frac{f\left(2\right)}{f\left(1\right)}+\frac{f\left(3\right)}{f\left(2\right)}+\frac{f\left(4\right)}{f\left(3\right)}+\frac{f\left(5\right)}{f\left(4\right)}=1+1+1+1=4$$
$$Sum\ of\ \frac{f\left(2\right)}{f\left(1\right)}to\ \frac{f\left(10\right)}{f\left(9\right)}=>$$
$$\frac{f\left(2\right)}{f\left(1\right)}+\frac{f\left(3\right)}{f\left(2\right)}+\frac{f\left(4\right)}{f\left(3\right)}+\frac{f\left(5\right)}{f\left(4\right)}+\frac{f\left(6\right)}{f\left(5\right)}+\frac{f\left(7\right)}{f\left(6\right)}+\frac{f\left(8\right)}{f\left(7\right)}+\frac{f\left(9\right)}{f\left(8\right)}+\frac{f\left(10\right)}{f\left(9\right)}=$$
$$1+1+1+1+1+1+1+1+1=9$$
$$Sum\ of\ \frac{f\left(2\right)}{f\left(1\right)}to\ \frac{f\left(n+1\right)}{f\left(n\right)}=>n$$
$$i.e\ \frac{f\left(2\right)}{f\left(1\right)}+\frac{f\left(3\right)}{f\left(2\right)}+...+\ \frac{f\left(n+1\right)}{f\left(n\right)}=>n$$
If n=4, we have;
$$\frac{f\left(2\right)}{f\left(1\right)}+\frac{f\left(3\right)}{f\left(2\right)}+\frac{f\left(4\right)}{f\left(3\right)}+\frac{f\left(5\right)}{f\left(4\right)}=4$$
If n=2005, we have;
$$\frac{f\left(2\right)}{f\left(1\right)}+\frac{f\left(3\right)}{f\left(2\right)}+\frac{f\left(4\right)}{f\left(3\right)}+\frac{f\left(5\right)}{f\left(4\right)}+\frac{f\left(2006\right)}{f\left(2005\right)}=2005$$.

Conclsuively, both statements combined together ARE SUFFICIENT.
Answer = option C

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by Max@Math Revolution » Sun Oct 06, 2019 5:52 pm
=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

Since we have many variables to determine a function and 0 equations, E is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
Since we have f(2)=f(1+1)=f(1)f(1), we have f(2)/f(1) = f(1)f(1)/f(1) = f(1)
Since we have f(3)=f(2+1)=f(2)f(1), we have f(3)/f(2) = f(2)f(1)/f(2) = f(1)
Since we have f(4)=f(3+1)=f(3)f(1), we have f(4)/f(3) = f(3)f(1)/f(3) = f(1)
...
Since we have f(2006)=f(2005+1)=f(2005)f(1), we have f(2006)/f(2005) = f(1)

So, we have f(2)/f(1) + f(3)/f(2) + f(4)/f(3) + ... +f(2006)/f(2005) = f(1) + f(1) + ... + f(1) = 2005*f(1) = 2005.

Both conditions together yield a unique solution, so they are sufficient.

Therefore, C is the answer.
Answer: C
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