Everything Maths Grade 12

CHAPTER 3. SEQUENCES AND SERIES 3.9

Therefore,

Sn =

a 1 (rn− 1)

r− 1

S∞ =

a 1 (0− 1)

r− 1

for− 1 < r < 1

=

−a 1

r− 1

=

a 1

1 −r

The sum of an infinite geometric series is givenby the formula

S∞=

�∞

i=1

a 1 ri−^1 =

a 1

1 −r

for− 1 < r < 1 (3.31)

where a 1 is the first term of the series and r is the common ratio.

See video: VMgly at http://www.everythingmaths.co.za

Exercise 3 - 5

1. What does (^25 )napproach as n tends towards∞?


2. Given the geometricseries:
   2. (5)^5 + 2. (5)^4 + 2. (5)^3 +...
   (a) Show that the seriesconverges
   (b) Calculate the sum toinfinity of the series
   (c) Calculate the sum ofthe first eight terms of the series, correct to two decimal places.
   (d) Determine
   �∞

n=9

2. 56 −n

correct to two decimal places using previously calculated results.

3. Find the sum to infinity of the geometric series 3 + 1 +^13 +^19 +...


4. Determine for whichvalues of x, the geometric series
   2 +^23 (x + 1) +^29 (x + 1)^2 +...
   will converge.


5. The sum to infinity of a geometric series withpositive terms is 416 and the sum of the firsttwo
   terms is 223. Find a, the first term, and r, the common ratio between consecutive terms.

More practice video solutions or help at http://www.everythingmaths.co.za

(1.) 01cz (2.) 01d0 (3.) 01d1 (4.) 01d2 (5.) 01d3

Chapter 3 End of Chapter Exercises