The fifth term of a geometric sequence is 32 and the eighth term is 4. (a) Find the ratio. (2 marks)
(b) Find the first term. (2 marks)
(C) Find the sum of the first 50 terms of the sequence. (3 marks)
What is the first term?

a. 1/2

b. 512

c. 1024

Step-by-step explanation:

The nth term of a geometric progression can be written as;

Tn = ar^n-1

where a is the first term, r is the common ratio and n is the term number

From the question;

Fifth term = ar^4 = 32

Eight term = ar^7 = 4

a. We want to find the common ratio

What to do here is that we can divide the 8th by the 5th term

ar^7/ar^4 = 4/32

r^3 = 1/8

r^3 = (1/2)^3

Thus r = 1/2

b. First term

we can make a substitution in any of the two given terms

Let’s use the fifth term

ar^4 = 32

a * (1/2)^4 = 32

a * 1/16 = 32

a = 16 * 32

a = 512

C. Sum of the first 50 terms

To calculate this, we use the sum of terms formula

Sn = a(1- r^n)/1-r

a = 512 , n = 50 and r = 1/2

Making the substitution;

Sn = 512(1-(1/2)^50)/1-1/2

Kindly note that (1/2)^50 will be infinitesimally small and will be close to zero

So we can approximate (1/2)^50 to be zero

Hence;

Sn = 512(1-0)/0.5

Sn = 512/0.5

Sn = 1024

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step-by-step explanation: