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

Let term be $a-2d,a-d,a,a+d,a+2d$a−2d,a−d,a,a+d,a+2d

Sum $=25\Rightarrow 5a=25\Rightarrow a=5$=25⇒5a=25⇒a=5

Product $=2520$=2520

$$(5-2d)(5-d)5(5+d)(5+2d)=2520$$(5−2d)(5−d)5(5+d)(5+2d)=2520

$$\Rightarrow (25-4{d}^{2})(25-{d}^{2})=504$$⇒(25−4d2)(25−d2)=504

$$\Rightarrow 625-100{d}^{2}-25{d}^{2}+4{d}^{4}=504$$⇒625−100d2−25d2+4d4=504

$$\Rightarrow 4{d}^{4}-125{d}^{2}+625-504=0$$⇒4d4−125d2+625−504=0

$$\Rightarrow 4{d}^{4}-125{d}^{2}+121=0$$⇒4d4−125d2+121=0

$$\Rightarrow 4{d}^{4}-121{d}^{2}-4{d}^{2}+121=0$$⇒4d4−121d2−4d2+121=0

$$\Rightarrow ({d}^{2}-1)(4{d}^{2}-121)=0$$⇒(d2−1)(4d2−121)=0

$$\Rightarrow d=\pm 1,d=\mp \frac{11}{2}$$⇒d=±1,d=∓211​

$d=\pm 1,$d=±1, does not give $-\frac{1}{2}$−21​ as a term

$$\therefore d=\frac{11}{2}$$∴d=211​

$\therefore$∴ Largest term $=5+2d=5+11=16$=5+2d=5+11=16