Monday, May 6, 2013

59-09 Basics of Arithmetic & Geometric Progression (Grades 9 to 12) Part-09

Note: 
1) To find three consecutive terms in GP, consider the terms as (a/r),  (a), (ar).
2) To find four consecutive terms in GP, consider the terms as (a/r3),  (a/r), (ar), (ar3).
3) To find five consecutive terms in GP, consider the terms as (a/r2),  (a/r), (a), (ar), (ar2).

A) Find three consecutive three terms in GP if the sum of the first and the third term is 35 and the product of all the three terms is 2744.

Solution:
1) Let " a " be the first term and " r " be the common ratio of GP.
2) Let three consecutive terms in GP be  (a/r),  (a), (ar).
3) According to the problem, 
                 (a/r) (a)(ar) = 2744
                             (a3) = 4 x 686
                             (a3) = 4 x 2 x 343
                             (a3) = 4 x 2 x 7 x 49
                             (a3) = (2 x 7)3
                               (a) = (2 x 7)
                               (a) = (14)
4) Simillarly,
                        a/r + ar = 35
                     a (1/r + r) = 35
                   14 (1/r + r) = 35
                        (1/r + r) = 35/14
                        (1/r + r) = 5/2
                        (1/r + r) = 2 + 1/2
5) This shows that r = 2 or 1/2
6) Answer: Taking r = 2 and a = 14, the three consecutive terms are 7, 14 and 28.
                    Taking r = 1/2 and a = 14, the three consecutive terms are 28, 14 and 7.
Note:
1) Arithmetic Mean: The three terms p, q and r in AP gives " q " as the arithmetic mean between p and q. So,
                 q - p = r - q
                 q + q = r + p
                    2 q = r + p
                       q = (r + p) / 2
2) Geometric Mean: The three terms p, q and r in GP gives " q " as the geometric mean between p and r. So,
                 q / p = r / q
                 q x q = r x p
                     q2 = r x p
                       q = (r p)1/2
Some Important Problems on AP & GP

A) In an AP  6, 12, 18, ....., how many terms shows the sum as 7650.

Solution:
1)   Here, a = 6, d = 6 and Sn = 7650.
2)   We know that 

        






















3) Answer: Total number of term in AP having sum as 7650 is 50.

Anil Satpute