“Leaders Are Anticipatory Thinkers. They Consider All Consequences Of Their Behaviors Before They Act.”

# Lecture - 13 Chapter 9 Sequences & Series

MISCELLANEOUS EXERCISE |

**Question 18. **If *a* and *b* are the roots of \( x^2-3x+p=0\) and *c*, *d* are roots of \(x^2-12x+q=0\), where *a*, *b*, *c*, *d* form a G.P. Prove that \((q+p):(q-p)=17:15\).

**Question 19. **The ratio of the A.M. and G.M. of two positive numbers a and b, is *m* : *n*. Show that \( a:b = (m+\sqrt{m^2-n^2}):(m-\sqrt{m^2-n^2})\).

**Question 20. **If *a*, *b*, *c* are in A.P.; *b*, *c*, *d* are in G.P. and \(\frac{1}{c}, \frac{1}{d}, \frac{1}{e}\) are in A.P. prove that *a*, *c*, *e* are in G.P.

**Question 21. **Find the sum of the following series up to n terms:

(i) 5 + 55 +555 + …

(ii) .6 +. 66 +. 666+…

**Question 22. **Find the 20^{th} term of the series 2 × 4 + 4 × 6 + 6 × 8 + … + *n* terms.

**Question 23. **Find the sum of the first *n* terms of the series: 3+ 7 +13 +21 +31 +…

**Question 24. **If \(S_1, S_2, S_3\) are the sum of first n natural numbers, their squares and their cubes, respectively, show that \(9{S_2}^2=S_3(1+8S_1)\).