Find the sum of the equation

Question

Find the sum.

(a)    1
3
+ 2
32
+ 22
33
+ 23
34
+    + 29
310

S = 

 

(b)    1 + 1

21/2

+ 1

2

+ 1

23/2

 +   

S = 

 

Answer 1

The first sequence is:

2⁰·3¹ + 2¹·3² + 2²·3³ + ... + 2⁹·3¹⁰

Each term is of the form 2^n-1·3ⁿ. You can write this as 2^n-1·3^n-1·3 = 3·6^n-1

which is a standard geometric sequence. Use the summation formula for a geometric sequence to find the answer.

Sum = a·(1-rⁿ) / (1-r)

where a = 3 and r = 6 and n = 10.

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The second sequence has the form:

1·2^(n-1)/2

which is geometric. Use the geometric summation formula given above to find the sum when a = 1, r = 2, n = 4.