Which term of the A.P. \[\mathbf{53},\text{ }\mathbf{48},\text{ }\mathbf{43},\ldots \] is the first negative term ?

Which term of the A.P.

$\mathbf{53},\text{ }\mathbf{48},\text{ }\mathbf{43},\ldots$

is the first negative term ?

Which term of the A.P.

$\mathbf{53},\text{ }\mathbf{48},\text{ }\mathbf{43},\ldots$

is the first negative term ?

From the question,

The first term a =

$53$

Then, difference d

$\begin{array}{*{35}{l}} =\text{ }48\text{ }\text{ }53\text{ }=\text{ }-5 \\ =\text{ }43\text{ }\text{ }48\text{ }=\text{ }-5 \\ \end{array}$

Therefore, common difference d =

$-5$

$\begin{array}{*{35}{l}} {{T}_{n}}~=\text{ }a\text{ }+\text{ }\left( n\text{ }\text{ }1 \right)d \\ =\text{ }53\text{ }+\text{ }\left( n\text{ }\text{ }1 \right)\left( -5 \right) \\ =\text{ }53\text{ }\text{ }5n\text{ }+\text{ }5 \\ =\text{ }58\text{ }\text{ }5n \\ 5n\text{ }=\text{ }58 \\ n=\text{ }58/5 \\ n\text{ }=\text{ }11.6\text{ }\approx \text{ }12 \\ \end{array}$

Therefore,

${{12}^{th}}$

term is the first negative term of the A.P.

$\mathbf{53},\text{ }\mathbf{48},\text{ }\mathbf{43},\ldots$

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