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In a GP, if the 3rd term is 27 and the 5th term is 243, what is the first term?

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Question: In a GP, if the 3rd term is 27 and the 5th term is 243, what is the first term?

Options:

  1. 3
  2. 9
  3. 1
  4. 27

Correct Answer: 9

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.

In a GP, if the 3rd term is 27 and the 5th term is 243, what is the first term?

Practice Questions

Q1
In a GP, if the 3rd term is 27 and the 5th term is 243, what is the first term?
  1. 3
  2. 9
  3. 1
  4. 27

Questions & Step-by-Step Solutions

In a GP, if the 3rd term is 27 and the 5th term is 243, what is the first term?
  • 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.
  • Step 11: Calculate the result: a = 3.
  • Geometric Progression (GP) – Understanding the properties of geometric sequences, including how to express terms in terms of the first term and common ratio.
  • Solving Equations – Using algebraic manipulation to solve for unknowns by setting up equations based on given terms.
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