1

### IIT-JEE 2006

Subjective
If $${a_n} = {3 \over 4} - {\left( {{3 \over 4}} \right)^2} + {\left( {{3 \over 4}} \right)^3} + ....{( - 1)^{n - 1}}{\left( {{3 \over 4}} \right)^n}\,\,and\,\,{b_n} = 1 - {a_n},$$, then find the least natural number $${n_0}$$ such that $${b_n}\,\, > \,\,{a_n}\,\forall \,n\,\, \ge \,\,{n_0}$$.

## Answer

6
2

### IIT-JEE 2003

Subjective
If a, b, c are in A.P., $${a^2}$$, $${b^2}$$, $${c^2}$$ are in H.P., then prove that either a = b = c or a, b, $${ - {c \over 2}}$$ form a G.P.

## Answer

solve it
3

### IIT-JEE 2002

Subjective
Let a, b be positive real numbers. If a, $${{A_1},{A_2}}$$, b are in arithmetic progression, a, $${{G_1},{G_2}}$$, b are in geometric progression and a, $${{H_1},{H_2}}$$, b are in harmonic progression, show that $$\,{{{G_1},{G_2}} \over {{H_1},{H_2}}} = {{{A_1} + {A_2}} \over {{H_1} + {H_2}}} = {{(2a + b)\,(a + 2b)} \over {9ab}}$$.

## Answer

solve it.
4

### IIT-JEE 2001

Subjective
Let $${a_1}$$, $${a_2}$$,.....,$${a_n}$$ be positive real numbers in geometric progression. For each n, let $${A_n}$$, $${G_n}$$, $${H_n}$$ be respectively, the arithmetic mean , geometric mean, and harmonic mean of $${a_1}$$,$${a_2}$$......,$${a_n}$$. Find an expression for the geometric mean of $${G_1}$$,$${G_2}$$,.....,$${G_n}$$ in terms of $${A_1}$$,$${A_2}$$,.....,$${A_n}$$,$${H_n}$$,$${H_1}$$,$${H_2}$$,........,$${H_n}$$.

## Answer

$$G = {\left( {{A_1},{A_2}....{A_n}\,{H_1},\,{H_2}.....{H_n}} \right)^{{\raise0.5ex\hbox{\scriptstyle 1} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{\scriptstyle {2n}}}}}$$

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