If the numbers $a, b, c, d, e$ form an A.P. then find values of $a - 4 d + 6 c - 4 d + e$ .

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Solution

Let $D$ be the common difference of the given A.P. Then,

$b = a + D, c = a + 2 D, d = a + 3 D$ and $e = a + 4 D$

$∴ a - 4 b + 6 c - 4 d + e = a - 4 (a + D) + 6 (a + 2 D) + (a + 3 D) + (a + 4 D)$

$⟹ a - 4 b + 6 c - 4 d + e = a - 4 a - 4 D + 6 a + 12 D - 4 a - 12 D + a + 4 D$

$⟹ a - 4 b + e = a - 4 a + 6 a = 4 a + a - 4 D + 12 D - 4 a - 12 D + a + 4 D$

$⟹ a - 4 b + 6 a - 4 d + e = 0$

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