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Question Number 15671 by tawa tawa last updated on 12/Jun/17

Prove by mathematcal induction that

1 + \frac{1}{1 + 2} + \frac{1}{1 + 2 + 3} + \dots + \frac{1}{1 + 2 + 3 + \dots n} = \frac{2 n}{n + 1}

Answered by icyfalcon999 last updated on 12/Jun/17

1) proving that the statement true when n = 1

R . H . S . = \frac{2 (1)}{1 + 1} = \frac{2}{2} = 1 = L . H . S .

2) suppose that the statement is true when n = k, k \in N u

1 + \frac{1}{1 + 2} + \frac{1}{1 + 2 + 3} + \dots + \frac{1}{1 + 2 + 3 + \dots k} = \frac{2 k}{k + 1}

3) proving that the statement true when n = k + 1

1 + \frac{1}{1 + 2} + \frac{1}{1 + 2 + 3} + \dots + \frac{1}{1 + 2 + 3 + \dots k} + \frac{1}{1 + 2 + 3 + \dots + k + 1} = \frac{2 (k + 1)}{k + 2}

L . H . S . = \frac{2 k}{k + 1} + \frac{1}{1 + 2 + 3 + \dots + k + 1}

= \frac{2 k}{k + 1} + \frac{1}{\frac{(k + 1) (k + 2)}{2}}

= \frac{2 k}{k + 1} + \frac{2}{(k + 1) (k + 2)}

= \frac{2 k (k + 2) + 2}{(k + 1) (k + 2)}

= \frac{{2 k}^{2} + 4 k + 2}{(k + 1) (k + 2)}

= \frac{2 (k^{2} + 2 k + 1)}{(k + 1) (k + 2)}

= \frac{2 {(k + 1)}^{2}}{(k + 1) (k + 2)}

= \frac{2 (k + 1)}{(k + 2)}

= R . H . S .

from 1, 2, 3 the statment is true for all natural numbers