Confused Candidate : Powers & Symmetric Expressions : If a+1/a=4 then what is the value of a4+1/a4 = ?

---

Rule: If x+1/x = n then what is the value of x²+1/x² = ?, Trick : n²-2

Rule: If 2x+1/2x = n then what is the value of 4x²+1/4x² = ?, Trick : n²-2

Rule: If 2x+1/3x = n then what is the value of = 4x²+1/9x² = ?, Trick : n²-2*2*1/3

Q. What is Integer ?

Definition: An integer is a whole number which can be positive, negative, or zero, without any fractional or decimal part. Integer বুলিলে সেই পূৰ্ণসংখ্যাক বুজায় যি ধনাত্মক, ঋণাত্মক বা শূন্য হ’ব পাৰে, আৰু ইয়াত কোনো ভগ্নাংশ বা দশমিক অংশ নাথাকে। Ex / উদাহৰণ:: …, -3, -2, -1, 0, 1, 2, 3, …

Note: 0 is an integer. Fractions (1/2, 3/4) are not integers . Decimals (2.5, -1.2) are not integers

Trick: Whenever a-1/a = x, (x is integer), the value of aⁿ-1/aⁿ must be divisible by x.

Q1. If x+1/x = 3 then what is the value of x²+1/x² = ?

Trick : n²-2

= n²-2 = 3²-2 = 9 - 2 = 7

Q2. If 2x+1/2x = 3 then what is the value of 4x²+1/4x² = ?

Trick : n²-2

= n² - 2 = 3² - 2 = 7

Q3. If 2x+1/3x = 3 then what is the value of = 4x²+1/9x² = ?

       Trick : n²-2*2*1/3

= n² - 2*2*1/3 = 3² - 2*2*1/3 = 9 - 4/3 = 23/3

Q4. If a+1/a=4 then what is the value of a⁴+1/a⁴ = ?

Options: (a) 204 (b) 196 (c) 198 (d) 194

Sol^n.

^{Trick : n2-2}

Square the given expression

a²+1/a²​=(a+1/a)²−2

= 4²−2 =16−2 =14

Again square

a⁴+1/a⁴​ = (a² + 1/a²)2 − 2

=14²-2=196−2=194

Ans: (d) 194

Q.  If a - 1/a = 7 then what the value of a⁵ - 1/a⁵ = ?

Options: (a) 18557 (b) 17147 (c) 14557 (d) 21107

Trick: Whenever a-1/a = x, (x is integer), the value of aⁿ-1/aⁿ must be divisible by x.

• Option (a) 18557 - divide by 7


• 18557 ÷ 7 = 2651 integer - correct


• If it is not divisible by 7, the answer is wrong.

Ans : Option (a) 18557