Example ① Find a and r when the sum of the geometric sequence a, ar, ar², ar³ is 60, and the sum of only the even-numbered terms, a, ar², ar⁴, …, is 24.
Solution: S=a(1+r+r²+r³)=60、T=a(1+r²)=24 (the sum of the odd-numbered terms among the four terms is calculated separately)…
※ The detailed conditions vary depending on the problem, so knowing which formula to use is key.
Solution: S=a(1+r+r²+r³)=60、T=a(1+r²)=24 (the sum of the odd-numbered terms among the four terms is calculated separately)…
※ The detailed conditions vary depending on the problem, so knowing which formula to use is key.
Key Point
- Use the number of terms and the given equations to set up a system of equations in a and r.
Practice Problems
- Find the number of terms n when the first term is 2, the common ratio is 3, and the sum is 242.
Answer and Explanation
- Answer:n = 5
2(3ⁿ−1)/2=3ⁿ−1=242 → 3ⁿ=243=3⁵。