Five numbers are in A.P., whose sum is 25 and product is 2520 . If one of these five numbers is - 1/2, then the greatest number amongst them is

(a): Let terms of the A.P. are a-2 d, a-d, a, \ a+d, a+2 d. Given, sum of terms =25 => 5 a=25 => a=5 Also, product of terms = 2520[Given] \ \ => (5-2 d)(5-d) 5(5+d)(5+2 d)=2520 => (25-4 d^2)(25-d^2)=504 => 625-100 d^2-25 d^2+4 d^4=504 => 4 d^4-125 d^2+121=0 \ => 4 d^4-121 d^2-4 d^2+121=0 => (d^2-1)(4 d^2-121)=0 \ => d= ± 1, d= ± (11 / 2)\ Since, d= ± 1 does not give (-1 / 2) as one of the term of the A.P., so d= ± (11 / 2) ∴ The required A.P. is -6,-(1 / 2), 5, (21 / 2), 16. Thus, greatest term is 16 .

Five numbers are in A.P., whose sum is 25 and product i... | Sequence & Series (Mathematics)

(a): Let terms of the A.P. are $a-2 d, a-d, a, \\ a+d, a+2 d$ . $\\$ Given, sum of terms $=25 \Rightarrow 5 a=25 \Rightarrow a=5$ $\\$ Also, product of terms = 2520[Given] $\\ \begin{aligned}& \Rightarrow(5-2 d)(5-d) 5(5+d)(5+2 d)=2520 \\& \Rightarrow\left(25-4 d^2\right)\left(25-d^2\right)=504 \\& \Rightarrow 625-100 d^2-25 d^2+4 d^4=504 \\& \Rightarrow 4 d^4-125 d^2+121=0 \\ & \Rightarrow 4 d^4-121 d^2-4 d^2+121=0 \\& \Rightarrow\left(d^2-1\right)\left(4 d^2-121\right)=0 \\ & \Rightarrow d= \pm 1, d= \pm \frac{11}{2}\end{aligned}$ $\\$ Since, $d= \pm 1$ does not give $\frac{-1}{2}$ as one of the term of the $\\$ A.P., so $d= \pm \frac{11}{2}$ $\\$ ∴ The required A.P. is $-6,-\frac{1}{2}, 5, \frac{21}{2}, 16$ . $\\$ Thus, greatest term is 16 .

Explanation

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