SEQUENCE AND SERIES (II)

Nth term of a G.P

A Geometric Progression is a sequence generated by multiplying or dividing a preceding term by a constant number to get a term. This constant number is called the common ratio designated by the letter r.

Evaluation

(1)        Find T9 of the G.P 5, 2½, 1¼, 5/8 …………..

(2)        If 3, p, q, 24 are consecutive term of an exponential sequence, find the values of  p and q.

Geometric Mean

Suppose x, y, z are consecutive terms of a geometric progression, then the common ratio r can be written as:

r = y/x = z/y

\y/x = z/y

y2 = xz

Evaluation

(1)        Insert two geometric mean between  -3 and –8/9.

(2)        The 4th term of a G.P in 75 and the 6th term is 192. Find the common ratio and the first term of the G.P

Sum to Infinity

The sum of the n terms as n approaches infinity is called the sum to infinity of the series and is designated S¥

Thus:

S¥ = a    if r<1

1-r

S¥ = a     if r>1

Examples:

Find the sum to infinity of the sequence 1, ¼, 1/161/64.

Solution

a = 1, r = ¼

S¥ =      1       =    1

1 – ¼       ¾

S¥ = 4/3

 

Evaluation

1.    The second and fourth terms of a G.P are 21 and 189. Find the sum of the first seven terms.

2.    Find the sum to infinity of 1+1/3 + 1/9 + 1/27 …………

GENERAL  EVALUATION

1.   Find the (a)sum of the first 8 terms  (b)sum to infinity of the series:  -5 , 5/2,  -5/4 , 5/8…….

2.   The sum of the first two terms of a G.P is 2½ and the sum of the first four terms is 311/18.

Find    the G.P if r > 0.

3.   Solve the following exponential equations  (a) 22x -6(2x) + 8 = 0        (b) 22x+1 -5(2x) + 2 = 0

READING ASSIGNMENTFurther Mathematics Project Book 1(New third edition).Chapter 33-36 & 37-45

 

WEEKEND ASSIGNMENT

1.          The sum to infinity of a G.P is 60. If the first term of the series 12, find its second term of the series 12, find its second term. A.   9.6        B.    6.9        C.  12.6       D.  8.6

2.          A G.P has 6 terms. If the 3rd and 4th terms are 28 and -56 respectively, find the sum of the G. P.

A.   471         B.   -471            C.   – 147        D. -741

3.          Find the sum of the G.P 2 + 6 + 18 + 54 + …………1458.  A. 8216   B. 6218   C.1682    D. 2186

4.          The 8th term of a G.P is -7/32. Find its common ratio if its first term is 28.

A. ½     B.   -½      C.  –2/3       D.   3/2

5.          Given the geometric progression 5, 10, 20, 40, 80 …………….. find its nth term.

A.   2(5n+2)        B.    5(2n+1)          C.   5(2n-1)           D.  2(5n-1)

          THEORY

1.          The fifth term of a G.P is greater than the fourth term by 13½, and the fourth term is greater than the third by 9. Find (i)  the common ratio   (ii)  the first term

2.          The sum of the first two terms of an exponential sequence is 135 and the sum of the third and the fourth terms is 60. Given that the common ratio is positive, calculate

(i)         the common        (ii)       the limit of the sum of the first n terms as n becomes large

(iii)        the least number of terms for which the sum exceeds 240

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