Correct Answer - Option 3 : 8
Concept:
Arithmetic means:
- It is defined as the sum of the values of all observations divided by the number of observations and is usually denoted by X.
- In general, if there are N observations as X1, X2, X3.......XN, then the Arithmetic Mean is given by
\(\bar X = \frac{X_1 + X_2 + X_3 + .......X_n}{N}\)
For two positive real numbers a, b
\(AM = \frac{a + b}{2}\)
Geometric mean: It is defined as the nth root of the product of n numbers.
Consider, if x1, x2 …. Xn is the observation, then the G.M is defined as
\(GM = (X_1\times X_2\times X_3 ..........X_n)^{\frac{1}{n}}\)
For two positive real numbers a, b
\(GM = \sqrt {ab}\)
Relation b/w AM and GM
\(AM - GM = \frac{a + b}{2} - \sqrt{ab} = \frac{(\sqrt a)^2 - (\sqrt b )^2}{2} \)
⇒ AM - GM ≥ 0
⇒ AM ≥ GM
Calculation:
We know that, for a positive real number, AM ≥ GM, therefore
\(\frac{\frac{1}{a^5} + \frac{1}{a^4} + \frac{1}{a^3} + \frac{1}{a^3} + \frac{1}{a^3} + 1+ a^8 + a^{10} }{8}\) ≥ \((\frac{1}{a^5} \times \frac{1}{a^4} \times \frac{1}{a^3} \times \frac{1}{a^3} \times \frac{1}{a^3} \times 1 \times a^8 \times a^{10})^{\frac{1}{8}}\)
\(\Rightarrow \frac{\frac{1}{a^5} + \frac{1}{a^4} + \frac{1}{a^3} + \frac{1}{a^3} + \frac{1}{a^3} + 1+ a^8 + a^{10} }{8}\) ≥ 1
\(\Rightarrow \frac{1}{a^5} + \frac{1}{a^4} + \frac{1}{a^3} + \frac{1}{a^3} + \frac{1}{a^3} + 1+ a^8 + a^{10} \) ≥ 8
Hence, the minimum value of a given sum will be equal to 1.
In the sum of real numbers a-5, a-4, 3a-3, 1, a8 and a10, we think that 6 terms are given, which are
a-5, a-4, 3a-3, 1, a8 and a10 but
for the sum and product, we will consider 8 terms, which are
a-5, a-4, a-3, a-3, a-3, 1, a8 and a10 (as 3 term of a-3 given)
