If the first term of a geometric progression is x and the common ratio is 1/2, what is the sum of the first 5 terms?
Practice Questions
1 question
Q1
If the first term of a geometric progression is x and the common ratio is 1/2, what is the sum of the first 5 terms?
x
x/2
x/3
x(1 - (1/2)^5)/(1 - 1/2)
The sum of the first n terms of a GP is given by S_n = a(1 - r^n) / (1 - r). Here, S_5 = x(1 - (1/2)^5) / (1 - 1/2) = x(1 - 1/32) / (1/2) = x(31/32) * 2 = x(62/32).
Questions & Step-by-step Solutions
1 item
Q
Q: If the first term of a geometric progression is x and the common ratio is 1/2, what is the sum of the first 5 terms?
Solution: The sum of the first n terms of a GP is given by S_n = a(1 - r^n) / (1 - r). Here, S_5 = x(1 - (1/2)^5) / (1 - 1/2) = x(1 - 1/32) / (1/2) = x(31/32) * 2 = x(62/32).
Steps: 11
Step 1: Identify the first term of the geometric progression (GP), which is given as x.
Step 2: Identify the common ratio of the GP, which is given as 1/2.
Step 3: Recall the formula for the sum of the first n terms of a GP: S_n = a(1 - r^n) / (1 - r).
Step 4: Substitute the values into the formula. Here, a = x, r = 1/2, and n = 5.
Step 5: Write the formula for the sum of the first 5 terms: S_5 = x(1 - (1/2)^5) / (1 - 1/2).
Step 6: Calculate (1/2)^5, which equals 1/32.
Step 7: Substitute this value back into the equation: S_5 = x(1 - 1/32) / (1/2).
Step 8: Simplify the expression inside the parentheses: 1 - 1/32 = 31/32.
Step 9: Now the equation looks like this: S_5 = x(31/32) / (1/2).
Step 10: Dividing by (1/2) is the same as multiplying by 2, so S_5 = x(31/32) * 2.
Step 11: Simplify the multiplication: S_5 = x(62/32).
