Dr. J's Maths.com
Where the techniques of Maths
are explained in simple terms.

Sequences & Series - Geometric - Summation questions.
Test Yourself 1.


 

A random encounter for fun. 1. For a particular series (neither arithmetic or geometric), the sum of n terms can be expressed as Sn. (These are referred to as partial sums).

If S1 = 9, S2 = 25 and S3 = 50,

find the first three terms of the series.

Answer.The terms are 9, 16 and 25.

  
Finding the sum.    
 

 

  
Finding a term. 7. The sum of the first 6 terms of a geometric series is 2912. The common ratio is 3.

Find the third term in the series.

Answer.As T1 = 8, T3 = 72.

 8. The sum of the first five terms of a geometric series is 211. The common ratio is 2/3.

Find the 8th term in the series.

Answer.As T1 = 81, T8 = 128/27.
  9. The sum of the first 6 terms of a geometric series is 315 and the common ratio is 1/2 (or = 0.5).

Find the six terms.

Answer.The terms are 160, 80, 40, 20, 10 and 5.

  
     
     
Finding the number of terms. 15. How many terms of the series

2 + 6 + 18 + 54 ... are needed to obtain a sum of 728?

Answer.6 terms.

 16. What is the least number of terms of the series 7 + 14 + 28 + ... that are required to give a sum just greater than 1,000?

Answer.8 terms are required.
     
     
Miscellaneous - finding more than one value. 21. Find the sum of the geometric series

2 + 10 + 50 + ....+ 156250.

Answer.Sum is 195,312.

 22. The second term of a geometric series is 2 and the fifth term is 16.

Find the sum of the first six terms.

Answer.Sum is 31.
     
Challenging 23.(i) Show that the sum Sn of the first n terms of the geometric series

xn-1 + 5xn-2 + 52xn-3 + .. + 5n-2x + 5n-1

is .

(ii) Use the definition of a derivative to show that when f(x) = xn, then f'(5) = 5n-1× n