Un = Sn - S(n-1)

= kn² - 3n - ( k(n-1)² - 3(n-1) )

= kn² - 3n - ( k(n² - 2n + 1) - 3n + 3)

= kn² - 3n - kn² + 2kn - k + 3n - 3

= 2kn - k - 3


So U(n-1)
= 2k(n-1) - k - 3
= 2kn - 2k - k - 3
= 2kn - 3k - 3


Then, Un - U(n-1)

= 2kn - k - 3 - (2kn - 3k - 3)
= 2kn - k - 3 - 2kn + 3k + 3
= 2k

The difference between any term and previous term (or between any term and the following term) is always 2k, a constant. This is the constant difference d and so the sequence is an arithmetic one.

Un = 2kn - k - 3

If U2, U3 and U6 are consecutive terms in a G.P, then they have a common ratio.

r = U3/U2 = U6/U3

(2k(3) - k - 3) / (2k(2) - k - 3) = (2k(6) - k -3) / (2k(3) - k - 3)

(5k - 3) / (3k - 3) = (11k - 3) / (5k - 3)

Cross multiply,

(5k - 3)² = (3k - 3)(11k - 3)

25k² - 30k + 9 = 33k² - 42k + 9

8k² - 12k = 0

2k² - 3k = 0

k(2k - 3) = 0

k = 0 (rejected as it is given that k is a non-zero real constant)

or

2k = 3
k = 1.5

Thank you! I need help with part b :(

Let the first term of the convergent G.P be a and the common ratio be r.


The first three terms are a, ar, ar²

Their product = a(ar)(ar²) = a³r³ = 1000

(ar)³ = 1000
ar = ³√1000 = 10
a = 10/r (or r = 10/a)


Add 6 to the second term, you get ar + 6
Add 7 to the third term, you get ar² + 7

Since these are now consecutive terms of an A.P, they have a common difference.

Second term - first term = third term - second term


ar + 6 - a = ar² + 7 - (ar + 6)
ar² - 2ar + a - 5 = 0

Sub a = 10/r,

(10/r)r² - 2(10) + 10/r - 5 = 0
10r - 25 + 10/r = 0
2r - 5 + 2/r = 0

Multiply the equation by r,
2r² - 5r + 2 = 0
(2r - 1)(r - 2) = 0
2r = 1 or r = 2 (rejected as |r| < 1 for a convergent G.P)

r = ½

Then a = 10/½ = 20

So first term is 20 and common ratio is 5.

Or

factor out 1/r,

1/r (2r² - 5r + 2) = 0

2r² - 5r + 2 = 0 (since 1/r ≠ 0)

Do the same as before.



Alternatively you can solve for a first.

ar² - 2ar + a - 5 = 0
Sub r = 10/a

a(10/a)² - 2a(10/a) + a - 5 = 0
100/a - 20 + a - 5 = 0
a - 25 + 100/a = 0


Multiply by a, a² - 25a + 100 = 0
(a - 20)(a - 5) = 0
a = 20 or a = 5

Then r = 10/20 or r = 10/5

r = ½ or r = 2 (rejected for same reason. So a = 5 is also rejected)




Other ways to factorise for r:

①
(2r - 1)(1 - 2/r) = 0
2r = 1 or 1 = 2/r
r = ½ or r = 2 (rejected for same reason)

so r = ½
a = 10 / ½ = 20


②
(2 - 1/r)(r - 2) = 0

2 = 1/r or r = 2 (rejected for same reason)
r = ½
So a = 10/½ = 20


For a:

①
(a - 5)(1 - 20/a) = 0
a = 5 or 20/a = 1 →a = 20

Then r = 10/5 or r = 10/20
r = 2(rejected for same reason. So a = 5 is also rejected)

or r = ½

②
(1 - 5/a)(a - 20) = 0
5/a = 1 or a = 20
a = 5

r = 10/5 or r = 10/20
r = 2 (rejected) or r = ½

Thank u!