In a cricket tournament 16 teams participated. A sum of Rs.8000 is awarded among themselves as prize money. If the last place team is awarded Rs.275 and the award increases by the same amount for successive finishing places, how much amount will the first place team receive ?

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Hint – To solve this problem we have to consider the series as an AP whose sum, first term , number of terms is given, so use the formula of AP to get the common difference and last term.

As we are considering it as an AP.
Where number of teams participated is the number of terms, which is,
 n = 16 ……(i)

The last ranked team got Rs.275
So the first term,
a = 275 ……(ii)
The total money awarded is Rs.8000
Therefore the sum is,
${{\text{S}}_n} = 8000$……(iii)
We also know,
  {{\text{S}}_n} = \dfrac{n}{2}(a + l) \\
  {\text{where, }}l{\text{ = last term}} \\
On putting the respective values of variables in above equation we get,
$8000 = \dfrac{{16}}{2}(275 + l)$
Then we calculate the last term as,
  l = \dfrac{{8000 \times 2}}{{16}} - 275 \\
  l = 1000 - 275 \\
  l = 725 \\
So, the last term is 725.
Hence, the winning team will be awarded Rs.725 .

Note – Whenever you get to solve these types of problems , think that, can we convert this situation into a series , it may be an AP,GP,HP etc. Then after considering it as a series solve it further to get the problem solved. Here we have considered the series as an AP and many of the terms of an AP were given. We have to find a single term by using the formula of sum of an AP. Alternatively we can also use the formula ${{\text{S}}_{\text{n}}}{\text{ = }}\frac{{\text{n}}}{{\text{2}}}{\text{(2a + (n - 1)d)}}$ then we found d then again we have to use the formula of last term as $l = a + (n - 1)d$ but the last term part is itself present in the formula of sum. So we have used it directly and got the problem solved.
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