Sum of the series n C 1-2 - 5 n C 2-3 - 52 n C 3- upto n terms isA- n - 6n-1B- 6 nC- 6n-1D- n - 6n

The correct option is A n ·6^n 1Let S=^n C_1+2 · 5 ^n C_2+3 · 5^2 ^nC_3+⋯ upto n terms nbsp; =∑_r=1^n 5^r 1· r · ^nC_r nbsp; =∑_r=1^n5^r 1· n · ^n 1C_r 1 = ...

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Standard XII

Mathematics

Binomial Expression

Sum of the se...

Question

Sum of the series $^{n} C_{1} + 2 \cdot 5^{n} C_{2} + 3 \cdot 5^{2}^{n} C_{3} + \dots$ upto $n$ terms is

n \cdot 6^{n - 1}

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6^{n - 1}

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6^{n}

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n \cdot 6^{n}

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Solution

The correct option is A $n \cdot 6^{n - 1}$
Let $S =^{n} C_{1} + 2 \cdot 5^{n} C_{2} + 3 \cdot 5^{2}^{n} C_{3} + \dots$ upto $n$ terms
$= n \sum r = 1 5^{r - 1} \cdot r \cdot^{n} C_{r}$
$= n \sum r = 1 5^{r - 1} \cdot n \cdot^{n - 1} C_{r - 1}$
$= n n \sum r = 1^{n - 1} C_{r - 1} \cdot 5^{r - 1}$
$= n \cdot (1 + 5)^{n - 1} (∵ n \sum s = 0^{n} C_{s} \cdot a^{r} = (1 + a)^{n})$
$= n \cdot 6^{n - 1}$

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Binomial Expression

Standard XII Mathematics

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Sum of the series nC1+2⋅5 nC2+3⋅52 nC3+⋯ upto n terms is

The correct option is A n⋅6n−1Let S=nC1+2⋅5 nC2+3⋅52 nC3+⋯ upto n terms =n∑r=15r−1⋅r⋅ nCr =n∑r=15r−1⋅n⋅ n−1Cr−1 =nn∑r=1n−1Cr−1⋅5r−1 =n⋅(1+5)n−1 (∵n∑s=0nCs⋅ar=(1+a)n) =n⋅6n−1

Sum of the series $^{n} C_{1} + 2 \cdot 5^{n} C_{2} + 3 \cdot 5^{2}^{n} C_{3} + \dots$ upto $n$ terms is