The question is about investments of a person which is in an AP. Another application based question from arithmetic progressions. With some simple but very powerful ideas, one can cut down on a lot of working when it comes to progressions. For example, anchoring a progression around its middle term can be very useful. CAT Exam tests the idea of progression often in the CAT Quantitative Aptitude section and this could also be tested in DI LR section of the CAT Exam as a part of a puzzle.

Question 17: Ram invest different amounts during the year on shares. S_{1}, S_{2}, S_{3}â€¦â€¦â€¦.S_{m} are different sums of â€˜nâ€™ amounts invested in â€˜mâ€™ years. If the amounts invested during the years are in A.P whose first terms are 1,2,3â€¦..m and common difference are 1,3,5â€¦..,(2m-1) respectively then find the total amount invested by Ram in â€˜mâ€™ years.

- n(m+1)
- m+1
- \\frac{mn}{2}\\)(mn+1)
- cannot be determined

\\frac{mn}{2}\\)(mn+1)

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Clearly, \\frac{A}{Q}\\) we have

S_{1} = \\frac{n}{2}\\) * [2 * 1 + (n-1) * 1] [âˆµ a=1 d=1]

S_{2} =\\frac{n}{2}\\) * [2 * 2 + (n-1) * 3] [âˆµ a=2 d=3]

S_{3} = \\frac{n}{2}\\) * [2 * 3 + (n-1) * 5] [âˆµ a=3 d=5]

S_{m} = \\frac{n}{2}\\) * [2 * m + (n-1) * (2m-1)] [âˆµa=m d= (2m-1)]

âˆ´ (S_{1} + S_{2} + S_{3}â€¦â€¦+ S_{m}) = \\frac{n}{2}\\) * [2*{1+2+3+4â€¦+m} +(n-1)*{1+3+5â€¦.+(2m-1)}]

= \\frac{n}{2}\\) * [{2 * \\frac{m(m+1)}{2}\\)} + {(n-1) * \\frac{m}{2}\\)(1+ 2m-1)}] {âˆµð?‘†=\\frac{ð?‘›}{2}\\)[(ð?‘Ž+ð?‘™)]}

= \\frac{n}{2}\\) * [m(m+1) + m^{2}(n-1)] = \\frac{mn}{2}\\) [(m+1)+m(n-1)]

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

The question is **"Find the total amount invested by Ram in â€˜mâ€™ years."**

Choice C is the correct answer.

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