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Question
A man accepts a position with an initial salary of ₹5200 per month. It is understood that he will receive an automatic increase of ₹320 in the very next month and each month thereafter.
(i) Find his salary for the tenth month.
(ii) What is his total earnings during the first year?
Solution
We have,
the initial salary, a1 = ₹5200,
the salary of the second month, a2 = ₹5200 + ₹320 = ₹5520,
the salary of the third month, a3 = ₹5520 + ₹320 = ₹5840,
\[\text { As, } a_2 - a_1 = 5520 - 5200 = 320 \text { and } a_3 - a_2 = 5840 - 5520 = 320\]
\[i . e . a_2 - a_1 = a_3 - a_2 \]
\[\text { So, } a_1 , a_2 , a_3 , . . . \text { are in A . P } . \]
\[\text { Also, } a = 5200, d = 320\]
\[\left( i \right) a_{10} = a + \left( 10 - 1 \right)d\]
\[ = 5200 + 9 \times 320\]
\[ = 5200 + 2880\]
\[ = 8080\]
\[\text { So, the salary of the man for the tenth month is } ₹ 8, 080 . \]
\[\left( ii \right) S_{12} = \frac{12}{2}\left[ 2a + \left( 12 - 1 \right)d \right]\]
\[ = 6\left( 2 \times 5200 + 11 \times 320 \right)\]
\[ = 6\left( 10400 + 3520 \right)\]
\[ = 6 \times 13920\]
\[ = 83520\]
\[\text { So, the total earnings of the man during the first year is } ₹ 83, 520 .\]
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