The sixth term of an arithmetic progression is $3$ and the difference exceeds $0.5$ . At what value of the difference of the progression is the product of the first, the fourth and the fifth term of the progression the greatest?

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Solution

$a + 5 d = 3 ⟹ a = 3 - 5 d$
product $= (a) (a + 3 d) (a + 4 d) ⟹ - 10 d^{3} + 51 d^{2} - 72 d + 27$ has extreme value at $d = 1, 2.4$
Highest product is at $2.4 = 9.72$

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The sixth term of an arithmetic progression is 3 and the difference exceeds 0.5. At what value of the difference of the progression is the product of the first, the fourth and the fifth term of the progression the greatest?

a+5d=3⟹a=3−5dproduct =(a)(a+3d)(a+4d)⟹−10d3+51d2−72d+27 has extreme value at d=1,2.4Highest product is at 2.4=9.72

The sixth term of an arithmetic progression is $3$ and the difference exceeds $0.5$ . At what value of the difference of the progression is the product of the first, the fourth and the fifth term of the progression the greatest?

$a + 5 d = 3 ⟹ a = 3 - 5 d$
product $= (a) (a + 3 d) (a + 4 d) ⟹ - 10 d^{3} + 51 d^{2} - 72 d + 27$ has extreme value at $d = 1, 2.4$
Highest product is at $2.4 = 9.72$