In a GP, if the 3rd term is 27 and the 5th term is 243, what is the first term?
Practice Questions
1 question
Q1
In a GP, if the 3rd term is 27 and the 5th term is 243, what is the first term?
3
9
1
27
Let the first term be a and the common ratio be r. Then, 3rd term = ar^2 = 27 and 5th term = ar^4 = 243. Dividing gives r^2 = 9, so r = 3. Substituting back gives a = 3.
Questions & Step-by-step Solutions
1 item
Q
Q: In a GP, if the 3rd term is 27 and the 5th term is 243, what is the first term?
Solution: Let the first term be a and the common ratio be r. Then, 3rd term = ar^2 = 27 and 5th term = ar^4 = 243. Dividing gives r^2 = 9, so r = 3. Substituting back gives a = 3.
Steps: 11
Step 1: Identify the first term as 'a' and the common ratio as 'r'.
Step 2: Write the formula for the 3rd term of a GP: 3rd term = ar^2. We know this equals 27, so we have the equation ar^2 = 27.
Step 3: Write the formula for the 5th term of a GP: 5th term = ar^4. We know this equals 243, so we have the equation ar^4 = 243.
Step 4: Now we have two equations: ar^2 = 27 and ar^4 = 243.
Step 5: To eliminate 'a', divide the second equation by the first equation: (ar^4) / (ar^2) = 243 / 27.
Step 6: Simplifying the left side gives r^2, and simplifying the right side gives 9. So we have r^2 = 9.
Step 7: Take the square root of both sides to find r: r = 3.
Step 8: Now substitute r back into the first equation: ar^2 = 27. Replace r with 3: a(3^2) = 27.
Step 9: Simplify: a(9) = 27.
Step 10: Solve for 'a' by dividing both sides by 9: a = 27 / 9.
