MyQuestionIcon

1

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

Homogeneous Function

Let n ∈ N, Sn...

Question

Let $n \in N, S_{n} = 3 n \sum r = 0^{3 n} C_{r}$ and $T_{n} = n \sum r = 0^{3 n} C_{3 r},$ then $9 | S_{n} - 3 T_{n} |$ is equal to

Open in App

Solution

$S_{n} =^{3 n} C_{0} +^{3 n} C_{1} + . . . +^{3 n} C_{3 n} = 2^{3 n}$
$T_{n} =^{3 n} C_{0} +^{3 n} C_{3} + . . . +^{3 n} C_{3 n}$
${(1 + x)}^{3 n} = (^{3 n} C_{0} +^{3 n} C_{3} x^{3} + . . .)$
$+ x (^{3 n} C_{1} +^{3 n} C_{4} x 3 + . . .) + x^{2} (^{3 n} C_{2} +^{3 n} C_{2} x^{5} + . . . .)$
Put $x = 1, ω, ω^{2}$ , we get
$\Rightarrow 2^{3 n} + (- ω^{2})^{3 n} + (- ω)^{3 n} = 3 [^{3 n} C_{0} +^{3 n} C_{3} + . . . .]$
$\Rightarrow 2^{3 n} + (- 1)^{3 n} [ω^{6 n} + ω^{3 n}] = 3 [^{3 n} C_{0} +^{3 n} C_{3} + . . . .]$
$= 3 T_{n}$
$\Rightarrow | S_{n} - 3 T_{n} | = 2$

Suggest Corrections

thumbs-up

0

Similar questions

S_{n} = n \sum r = 0 \frac{1}{^{n} C_{r}}

t_{n} = n \sum r = 0 \frac{r}{^{n} C_{r}}

\frac{t_{n}}{S_{n}}

S_{n} = n \sum r = 0 \frac{1}{^{n} C_{r}}

t_{n} = n \sum r = 0 \frac{r}{^{n} C_{r}}

\frac{t_{n}}{s_{n}} =

t_{n} = \sum_{r = 0}^{n} \frac{1}{{({^{n} C}_{r})}^{k}}

S_{n} = \sum_{r = 0}^{n} \frac{r}{{({^{n} C}_{r})}^{k}}

k \in Z^{+}

{cos}^{- 1} (\frac{S_{n}}{n t_{n}})

n \in N

S (n) = \sum_{r = 0}^{n} {(- 1)}^{r} \frac{1}{({^{n} C}_{r})}

S = \sum_{r = 0}^{n} {(- 1)}^{r} \frac{({^{r + 2} C}_{r})}{({^{n} C}_{r})}

T_{n} = n \sum r = 1 \frac{n}{r^{2} - 2 r . n + 2 n^{2}}, S_{n} = n - 1 \sum r = 0 \frac{n}{r^{2} - 2 r . n + {2 n}^{2}}

View More

Join BYJU'S Learning Program

similar_icon

Related Videos

lock

Methods of Solving First Order, First Degree Differential Equations

MATHEMATICS

Watch in App

explore_more_icon

Explore more

Homogeneous Function

Standard XII Mathematics

Join BYJU'S Learning Program

CrossIcon