To find: Value of (2+√3)7 + (2-√3)7
Formula used: (i)
nCr = \(\frac{n!}{(n-r)!(r)!}\)
(ii) (a+b)n = nC0an + nC1an-1b + nC2an-2b2 + … +nCn-1abn-1 + nCnbn
⇒ 2[(1)a7 + (21)a5b2 + (35)a3b4 + (7)ab6]
⇒ 2[a7 + 21a5b2 + 35a3b4 + 7ab6] = (a+b)7 + (a-b)7
Putting the value of a = 2 and b = √3 in the above equation
(2+√3)7 + (2-√3)7
= 2[{27} + {21(2)5(√3)2} + {35(2)3(√3)4} + {7(2)(√3)6}]
= 2[128 + 21(32)(3)+ 35(8)(9) + 7(2)(27)]
= 2[128 + 2016 + 2520 + 378]
= 10084