Question-194593

Question Number 194593 by pascal889 last updated on 10/Jul/23

Answered by MM42 last updated on 10/Jul/23

u_{n} = u_{n - 1} + n + 1; n ⩾ 2, u_{1} = 4

u_{n} = u_{n - 1} + (n + 1)

= u_{n - 2} + (n) + (n + 1)

= u_{n - 3} + (n - 1) + (n) + (n + 1)

⋮

= u_{2} + 4 + 5 + \dots + (n) + (n + 1)

= u_{1} + 3 + 4 + \dots + (n) + (n + 1)

\Rightarrow u_{n} = 4 + \frac{(n - 1) (n + 4)}{2} = \frac{n^{2}}{2} + \frac{3 n}{2} + 2 ✓

Answered by Frix last updated on 10/Jul/23

4 + 3 = 7

7 + 4 = 11

11 + 5 = 16

u_{n} = \frac{n^{2}}{2} + \frac{3 n}{2} + 2

u_{100000} = 5000150002

Commented by BaliramKumar last updated on 10/Jul/23

please detail

how to calculate u_{n} = \frac{n^{2}}{2} + \frac{3 n}{2} + 2

Commented by Frix last updated on 10/Jul/23

u_{n} = c_{2} n^{2} + c_{1} n + c_{0}

Insert all given pairs (n, u_{n}) and solve the

system .

Answered by horsebrand11 last updated on 11/Jul/23

U_{n} = 4 + S_{n - 1}

U_{n} = 4 + \frac{n - 1}{2} (6 + (n - 2) . 1)

U_{n} = 4 + \frac{n - 1}{2} (n + 4)

U_{n} = \frac{1}{2} (n^{2} + 3 n + 4)