Mathematics
If Sn denotes the sum of first n terms of an A.P., prove that S30 = 3(S20 - S10).
AP GP
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Answer
By formula Sn =
We need to prove S30 = 3(S20 - S10).
{30} = \dfrac{30}{2}[2a + (30 - 1)d] \\[1em] = 15[2a + 29d] \\[1em] = 30a + 335d \\[1em] \Rightarrow \text{R.H.S.} = 3(S{20} - S_{10}) \\[1em] = 3\Big(\dfrac{20}{2}[2a + (20 - 1)d] - \dfrac{10}{2}[2a + (10 - 1)d]\Big) \\[1em] = 3\Big(10[2a + 19d] - 5[2a + 9d]\Big) \\[1em] = 3(20a + 190d - 10a - 45d) \\[1em] = 3(10a + 145d) \\[1em] = 30a + 335d.
∴ L.H.S. = R.H.S. = 30a + 335d.
Hence, proved that S30 = 3(S20 - S10).
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