Let Tr be the rth term of a sequence- for R-1- 2- 3- - If 3Tr-1-Tr and T7-1243- then the value of -r-1Tr- Tr-1 is-

The correct option is D 278 T_r+1T_r=13 If T_1=a ; T_2=a3 ; T_3=a3^2 , #160; ∴ T_7=a3^6=1243⇒a3^5× 3=13^5⇒ a=3 ∴ T_r· T_r+1= ...

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Standard X

Mathematics

Patterns and Sequences

Let Tr be t...

Question

Let $T_{r}$ be the $r^{t h}$ term of a sequence, for $R = 1, 2, 3$ ,.... . If $3 T_{r + 1} = T_{r}$ and $T_{7} = \frac{1}{243}$ , then the value of $\infty \sum r = 1 (T_{r} \cdot T_{r + 1})$ is?

\frac{9}{2}

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\frac{27}{8}

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\frac{81}{8}

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\frac{81}{4}

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Solution

The correct option is D $\frac{27}{8}$
$\frac{T_{r + 1}}{T_{r}} = \frac{1}{3}$
If $T_{1} = a$ ; $T_{2} = \frac{a}{3}$ ;
$T_{3} = \frac{a}{3^{2}}$ ,
$∴ T_{7} = \frac{a}{3^{6}} = \frac{1}{243} \Rightarrow \frac{a}{3^{5} \times 3} = \frac{1}{3^{5}} \Rightarrow a = 3$
$∴ T_{r} \cdot T_{r + 1} = \frac{a}{3^{r - 1}} \times \frac{a}{3^{r}} = \frac{a^{2}}{3^{2 r - 1}} = \frac{3^{2}}{3^{2 r - 1}}$
$\infty \sum r = 1 T_{r} \cdot T_{r + 1}$
$= (3 + \frac{1}{3} + \frac{1}{3^{3}} + \frac{1}{3^{5}} + . . . \infty)$
$= \frac{3}{1 - \frac{1}{3^{2}}} = \frac{27}{8}$

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Similar questions

Let $T_{r}$ be the $r^{t h}$ term of a sequence. If, for r = 1,2,3,.... . $3 T_{r + 1} = T_{r}$ and $T_{7} = \frac{1}{243},$ then the value of $\sum_{r = 1}^{\infty} (T_{r} . T_{r + 1})$ is

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equals?

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Let Tr be the rth term of a sequence, for R=1,2,3,.... . If 3Tr+1=Tr and T7=1243, then the value of ∞∑r=1(Tr⋅Tr+1) is?

The correct option is D 278Tr+1Tr=13If T1=a; T2=a3;T3=a32, ∴T7=a36=1243⇒a35×3=135⇒a=3∴Tr⋅Tr+1=a3r−1×a3r=a232r−1=3232r−1∞∑r=1Tr⋅Tr+1=(3+13+133+135+...∞)=31−132=278

Let $T_{r}$ be the $r^{t h}$ term of a sequence, for $R = 1, 2, 3$ ,.... . If $3 T_{r + 1} = T_{r}$ and $T_{7} = \frac{1}{243}$ , then the value of $\infty \sum r = 1 (T_{r} \cdot T_{r + 1})$ is?