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Question Number 10487 by Saham last updated on 13/Feb/17

A sequence of numbers T_{1}, T_{2}, T_{3}, \dots \dots . T_{n} satisfies

the relation T_{n + 1} + n^{2} = {nT}_{n} + 2 for all integers

n ⩾ 1 . if T_{1} = 2 . find

(a) The values of T_{2}, T_{3}, T_{4}

(b) An expression for T_{n} in terms of the sequence

(c) The sum of the first nth terms of the sequence

(d) The sum of T_{n} + T_{n + 1} + T_{n + 2} when n = 20

Answered by mrW1 last updated on 17/Feb/17

T_{n + 1} = {nT}_{n} + 2 - n^{2}

(a)

T_{1} = 2

T_{2} = 1 \times T_{1} + 2 - 1^{2} = 2 + 2 - 1 = 3

T_{3} = 2 \times T_{2} + 2 - 2^{2} = 6 + 2 - 4 = 4

T_{4} = 3 \times T_{3} + 2 - 3^{2} = 12 + 2 - 9 = 5

(b)

T_{n} = n + 1

(c)

S (n) = \underset{k = 1}{\sum^{n}} T_{k} = 2 + 3 + 4 + \dots + (n + 1) = \frac{n (n + 1)}{2}

(d)

T_{20} + T_{21} + T_{22} = 21 + 22 + 23 = 66

Commented by Saham last updated on 18/Feb/17

I really appreciate sir .

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