Ramkali required Rs. 2500 after 12 weeks to send her daughter to school. She saved Rs. 100 in the first week and increased her weekly savings by Rs. 20 every week. Find whether she will be able to send her daughter to school after 12 weeks.
ANSWER
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Hint: Find the series of money which is being saved every week. Use the formula of the sum of $n$ terms of an Arithmetic progression to determine the sum of saved money for 12 weeks. If the sum exceeds Rs. 2500 then she will be able to send her daughter to school otherwise not.
Complete step-by-step answer: First of all let us determine the money that is being saved every week. It is given that, in the first week she saves Rs. 100 and makes an increment of Rs. 20 every week. Therefore, the series that will be formed is: 100, 120, 140, 160,…………up to 12 weeks. We can see that this forms an A.P. with first term as Rs. 100 and common difference as Rs. 20. Now, the sum of money that is saved after 12 weeks will be equal to the sum of the 12 terms of this A.P. We know that sum of $n$ terms of an A.P is given by ${{S}_{n}}=\dfrac{n}{2}\left( 2a+(n-1)d \right)$. Therefore, substituting the value: $a=100,\text{ }n=12,\text{ }d=20$ in the above equation, we get, $\begin{align} & {{S}_{12}}=\dfrac{12}{2}\times \left( 2\times 100+(12-1)\times 20 \right) \\ & =6\times \left( 200+11\times 20 \right) \\ & =6\times \left( 200+220 \right) \\ & =6\times \left( 420 \right) \\ & =2520 \\ \end{align}$ Therefore, she will save Rs. 2520 at the end of 12 weeks. Hence, we can conclude that she will be able to send her daughter to school.
Note: One may get confused in understanding the words of this question and forming the sequence of the savings. One can notice that, it is said in the question ‘every week she is increasing the savings by Rs. 20’. This means that the difference between the savings of two consecutive weeks is Rs. 20 and this is constant. This gives us the hint that the sequence is an A.P.