Write the 50th term of the series 2 + 3 + 6 + 11 + 18 + ...

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#### Solution

We have,

\[ a_1 = 2, \]

\[ a_2 = 3 = 2 + 1, \]

\[ a_3 = 6 = 2 + 1 + 3, \]

\[ a_4 = 11 = 2 + 1 + 3 + 5, \]

\[ a_{50} = 2 + 1 + 3 + 5 + . . . \left( 50 \text { terms } \right)\]

\[ = 2 + \frac{49}{2}\left[ 2 \times 1 + \left( 49 - 1 \right) \times 2 \right] \left( \text { As, the terms apart 2 are in A . P . with a = 1 and d = 2 } \right)\]

\[ = 2 + \frac{49}{2}\left( 2 + 48 \times 2 \right)\]

\[ = 2 + \frac{49}{2} \times 98\]

\[ = 2 + {49}^2 \]

\[ = 2 + 2401\]

\[ = 2403\]

So, the 50th term of the given series is 2403.

Concept: Sum to N Terms of Special Series

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