Sequences and Series 5 Question 23
14. Let $S _k$, where $k=1,2,, \ldots, 100$, denotes the sum of the infinite geometric series whose first term is $\frac{k-1}{k !}$ and the common ratio is $\frac{1}{k}$. Then, the value of $\frac{100^{2}}{100 !}+\sum _{k=1}^{100}\left|\left(k^{2}-3 k+1\right) S _k\right|$ is …..
(2010)
Assertion and Reason
For the following question, choose the correct answer from the codes (a), (b), (c) and (d) defined as follows:
(a) Statement I is true, Statement II is also true; Statement II is the correct explanation of Statement I
(b) Statement I is true, Statement II is also true; Statement II is not the correct explanation of Statement I
(c) Statement I is true; Statement II is false
(d) Statement I is false; Statement II is true
Show Answer
Solution:
- Given, $a _1 a _2 a _3 \ldots a _n=c$
$\Rightarrow \quad a _1 a _2 a _3 \ldots\left(a _{n-1}\right)\left(2 a _n\right)=2 c$
$\therefore \quad \frac{a _1+a _2+a _3+\ldots+2 a _n}{n} \geq\left(a _1 \cdot a _2 \cdot a _3 \ldots 2 a _n\right)^{1 / n}$
[using $AM \geq GM$ ]
$\Rightarrow \quad a _1+a _2+a _3+\ldots+2 a _n \geq n(2 c)^{1 / n}$
[from Eq. (i)]
$\Rightarrow$ Minimum value of
$$ a _1+a _2+a _3+\ldots+2 a _n=n(2 c)^{1 / n} $$
