If 5 times the fifth term of an AP is equal to 8 times its eighth term, show that its 13th term is zero.
If 5 times the fifth term of an AP is equal to 8 times its eighth term, show that its 13th term is zero.
Solution and Explanation
The given arithmetic progression (AP) is represented as a₁, a₂, a₃, ..., a₁₃
We are given that: 5a₅ = 8a₈
Using the property of the arithmetic progression, we have: 5(a + 4d) = 8(a + 7d)
Expanding the expressions: 5a + 20d = 8a + 56d
Rearranging the terms: 3a + 36d = 0
Factoring out 3:
3(a + 12d) = 0
Since the term "a + 12d" is equal to zero, we have:
a + 12d = 0
And solving for the 13th term:
a₁₃ = a + (13 - 1)d a₁₃ = a + 12d
Substituting the value we obtained earlier: a₁₃ = 0
Hence, the 13th term of the arithmetic progression is 0.
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