Numerical answer
The sum of the first 10 terms of an arithmetic progression is 175 and the sum of its first 20 terms is 650. What is the 30th term of the progression?
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Answer: 91
S₁₀ = 10/2 × (2a + 9d) = 175 ⇒ 2a + 9d = 35.
S₂₀ = 20/2 × (2a + 19d) = 650 ⇒ 2a + 19d = 65.
Subtracting: 10d = 30 ⇒ d = 3; then 2a = 35 − 27 = 8 ⇒ a = 4.
30th term = a + 29d = 4 + 87 = 91.
Check: S₁₀ = 5(4 + 31) = 175 ✓.
S₂₀ = 20/2 × (2a + 19d) = 650 ⇒ 2a + 19d = 65.
Subtracting: 10d = 30 ⇒ d = 3; then 2a = 35 − 27 = 8 ⇒ a = 4.
30th term = a + 29d = 4 + 87 = 91.
Check: S₁₀ = 5(4 + 31) = 175 ✓.
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