A function f(x) is given by f(x)=5x5x+5, then the sum of the series: f120+f220+f320+......+f3920 is equal to: A192 B492 C392 D292

The ideal decision is C 392Explanation for the correct answer:Step 1: Finding f(2-x)The function is f(x)=5x5x+5.Finding the sum requires finding f(2-x)f(2-x)=52-x52-x+5=525x525x+5=55+5xStep 2: Add f(x),f(2-x)f(x)+f(2-x)=5x5x+5+55x+5=1Now consider, x=120⇒fx=f120⇒f(2-x)=f2-120=f3920Step 3:Finding the pairsf120+f3920=1∵f(x)+f(2-x)=1Similarly, the pairs aref220+f3820=1,....................f1920+f2120=1and f2020=f(1)f(1)=5151+5=12Step 4: Finding the summationf120+f220+f320+......+f3920There are 19 such pairs and f(1)Consequently, the series' total is⇒(19)(1)+12⇒392Hence, option (C) is the correct answer.

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High School
Geometry
Trigonometry
Trigonometric Functions

fluorekyz4l

Answered question

2023-02-26

A function

f (x)

is given by

f (x) = \frac{5^{x}}{5^{x} + 5}

, then the sum of the series:

f (\frac{1}{20}) + f (\frac{2}{20}) + f (\frac{3}{20}) + . . . . . . + f (\frac{39}{20})

is equal to: A

\frac{19}{2}

\frac{49}{2}

\frac{39}{2}

\frac{29}{2}

Answer & Explanation

Gustavo Gamble

Beginner2023-02-27Added 3 answers

The ideal decision is C

\frac{39}{2}

Explanation for the correct answer:
Step 1: Finding

f (2 - x)

The function is

f (x) = \frac{5^{x}}{5^{x} + 5}

.
Finding the sum requires finding

f (2 - x)

f (2 - x) = \frac{5^{2 - x}}{5^{2 - x} + 5} = \frac{\frac{5^{2}}{5^{x}}}{\frac{5^{2}}{5^{x}} + 5} = \frac{5}{5 + 5^{x}}

Step 2: Add

f (x), f (2 - x)

f (x) + f (2 - x) = \frac{5^{x}}{5^{x} + 5} + \frac{5}{5^{x} + 5} = 1

Now consider,

x = \frac{1}{20}

\Rightarrow f (x) = f (\frac{1}{20}) \Rightarrow f (2 - x) = f (2 - \frac{1}{20}) = f (\frac{39}{20})

Step 3:Finding the pairs

f (\frac{1}{20}) + f (\frac{39}{20}) = 1 (∵ f (x) + f (2 - x) = 1)

Similarly, the pairs are

f (\frac{2}{20}) + f (\frac{38}{20}) = 1, . . . . . . . . . . . . . . . . . . . . f (\frac{19}{20}) + f (\frac{21}{20}) = 1

and

f (\frac{20}{20}) = f (1) f (1) = \frac{5^{1}}{5^{1} + 5} = \frac{1}{2}

Step 4: Finding the summation

f (\frac{1}{20}) + f (\frac{2}{20}) + f (\frac{3}{20}) + . . . . . . + f (\frac{39}{20})

There are

19

such pairs and

f (1)

Consequently, the series' total is

\Rightarrow (19) (1) + \frac{1}{2} \Rightarrow \frac{39}{2}

Hence, option (C) is the correct answer.

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