If 2 f xy - f x y - f y x - x - y - R and f 1-3- then the value of - r -110 f r is equal toA- 3-2310-1- 3-239-1C- 1-2310-1D- 1-239-1

The correct option is A 32(3^10 1)Given, 2f(xy)=( f(x))^y+( f(y))^x and f(1)=3 nbsp; Put y=1 ⇒ 2f(x)=f(x)+( f(1))^x ⇒ f(x)=3^x ∴∑_r=1^10f(r) =∑_r=1^103^r=3+ ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Sum of n Terms

If 2 f xy = f...

Question

If $2 f (x y) = {(f (x))}^{y} + {(f (y))}^{x} \forall x, y \in R$ and $f (1) = 3,$ then the value of $10 \sum r = 1 f (r)$ is equal to

\frac{3}{2} (3^{10} - 1)

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

\frac{3}{2} (3^{9} - 1)

No worries! We‘ve got your back. Try BYJU‘S free classes today!

\frac{1}{2} (3^{10} - 1)

No worries! We‘ve got your back. Try BYJU‘S free classes today!

\frac{1}{2} (3^{9} - 1)

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is A $\frac{3}{2} (3^{10} - 1)$
Given, $2 f (x y) = {(f (x))}^{y} + {(f (y))}^{x}$ and $f (1) = 3$
Put $y = 1$
$\Rightarrow 2 f (x) = f (x) + {(f (1))}^{x}$
$\Rightarrow f (x) = 3^{x}$

$∴ 10 \sum r = 1 f (r)$
$= 10 \sum r = 1 3^{r} = 3 + 3^{2} + 3^{3} + \dots + 3^{10} = \frac{3}{2} (3^{10} - 1)$
$(∵$ For G.P., $S_{n} = \frac{a (r^{n} - 1)}{r - 1})$

Suggest Corrections

Similar questions

Q. Consider a real valued function

f (x)

satisfying

2 f (x y) = (f (x))^{y} + (f (y))^{x} \forall x, y \in R

and

f (1) = p

where

p \neq 1

then find

(p - 1) n \sum r = 1 f (r)

$f (x) = 2 x^{2} + b x + c$ and $f (0) = 3$ and $f (2) = 1$ , then $f (1)$ is equal to

Let

f (x) = \frac{1}{2} [f (x y) + f (\frac{x}{y})]

for

x, y \in R^{+}

such that

f (1) = 0; f^{'} (1) = 2

f^{'} (3)

is equal to

Q. If

f (n) = α^{n} + β^{n}

and

∣ ∣ ∣ ∣ ∣ \begin{matrix} 3 & 1 + f (1) & 1 + f (2) 1 + f (1) & 1 + f (2) & 1 + f (3) 1 + f (2) & 1 + f (3) & 1 + f (4) \end{matrix} ∣ ∣ ∣ ∣ ∣ = k (1 - α)^{2} (1 - β)^{2} (α - β)^{2}

, then k is equal to

If $y = | cos x | + | sin x |$ then $\frac{d y}{d x}$ at $x = \frac{2 π}{3}$ is:

Join BYJU'S Learning Program

If 2f(xy)=(f(x))y+(f(y))x ∀ x,y∈R and f(1)=3, then the value of 10∑r=1f(r) is equal to

The correct option is A 32(310−1)Given, 2f(xy)=(f(x))y+(f(y))x and f(1)=3 Put y=1 ⇒2f(x)=f(x)+(f(1))x ⇒f(x)=3x ∴10∑r=1f(r) =10∑r=13r=3+32+33+⋯+310=32(310−1) (∵ For G.P., Sn=a(rn−1)r−1)

If $2 f (x y) = {(f (x))}^{y} + {(f (y))}^{x} \forall x, y \in R$ and $f (1) = 3,$ then the value of $10 \sum r = 1 f (r)$ is equal to