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Explain Bot To find the 20th term of an arithmetic progression, we need to determine the common difference (d) and the first term (a₁) of the sequence.
Given that the 5th term is 0 and the 10th term is 10, we can use this information to find the common difference.
Step 1: Identify the terms:
a₅ = 0 (5th term)
a₁₀ = 10 (10th term)
Step 2: Use the formula for the n-th term of an arithmetic progression:
aₙ = a₁ + (n - 1) × d
Step 3: Plug the values of a₅ and a₁₀ into the formula:
0 = a₁ + (5 - 1) × d
10 = a₁ + (10 - 1) × d
Step 4: Simplify the equations:
Equation 1: a₁ + 4d = 0
Equation 2: a₁ + 9d = 10
Step 5: Solve the equations simultaneously:
Subtract Equation 1 from Equation 2:
(a₁ + 9d) - (a₁ + 4d) = 10 - 0
5d = 10
Divide both sides by 5:
d = 2
Step 6: Substitute the value of d into either Equation 1 or 2 to find a₁:
a₁ + 4(2) = 0
a₁ + 8 = 0
a₁ = -8
Therefore, the first term (a₁) is -8 and the common difference (d) is 2.
Step 7: Use the formula to find the 20th term:
a₂₀ = a₁ + (20 - 1) × d
a₂₀ = -8 + (19) × 2
a₂₀ = -8 + 38
a₂₀ = 30
Hence, the 20th term of the arithmetic progression is 30.