dear student LHS=∑r=0n-1log2r+2r+1=log221+log232+ +log2n+1n=log221×32× ×n+1n=log2(n+1)RHS=∏r=1099logr(r+1)=∏r=1099log(r+1)log(r)=log11log10×log12log11× ×log100log99=log100log10=2solog2(n+1)=2n+1=4n=3odd number regards

Q6 pl

dear student

L H S = \sum_{r = 0}^{n - 1} \log_{2} (\frac{r + 2}{r + 1}) = l {og}_{2} (\frac{2}{1}) + l {og}_{2} (\frac{3}{2}) + . . . . + l {og}_{2} (\frac{n + 1}{n}) = l {og}_{2} (\frac{2}{1} \times \frac{3}{2} \times . . . \times \frac{n + 1}{n}) = l {og}_{2} (n + 1) R H S = \prod_{r = 10}^{99} \log_{r} (r + 1) = \prod_{r = 10}^{99} \frac{\log (r + 1)}{\log (r)} = \frac{\log 11}{\log 10} \times \frac{\log 12}{\log 11} \times . . . \times \frac{\log 100}{\log 99} = \frac{\log 100}{\log 10} = 2 s o l {og}_{2} (n + 1) = 2 n + 1 = 4 n = 3 o d d n u m b e r

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