Question

if a+1/a=5 find a7-109a4-114a

Answer 1

Answer:

this is the answer of your question.

Answer 2

Given. a + 1/a = 5

To find. a⁷ - 109a⁴ - 114a

Solution.

Here, a + 1/a = 5

or, a² - 5a + 1 = 0 .....(1)

Using quadratic formula, we get

a = [- (- 5) ± √{(- 5)² - 4 * 1 * 1}]/2

= [5 ± √(25 - 4)]/2

= (5 ± √21)/2

When a = (5 + √21)/2,

a⁷ - 109a⁴ - 114a

= ((5 + √21)/2)⁷ - 109 ((5 + √21)/2)⁴

- 114 ((5 + √21)/2)

= - 24

When a = (5 - √21)/2,

a⁷ - 109a⁴ - 114a

= ((5 - √21)/2)⁷ - 109 ((5 - √21)/2)⁴

- 114 ((5 - √21)/2)

= - 24

Answer.

a⁷ - 109a⁴ - 114a = - 24

Another approach. Division

[ refer to the attachment ]

∴ a⁷ - 109a⁴ - 114a

= (a² - 5a + 1) (a⁵ + 5a⁴ + 24a³ + 6a²

+ 6a + 24) - 24

= 0 * (a⁵ + 5a⁴ + 24a³ + 6a² + 6a + 24) - 24

[ by (1) ]

= - 24

Remark. There can be an assumption of the problem: (a + 1)/a = 5. Find a and put in a⁷ - 109a⁴ - 114a.