Question

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 saving by Rs.20 every week. Find whether she will be able to send her daughter to school after 12 weeks. What value is generated in the above situation?

Answer 1

Hint: In this question, we need to determine the total amount accumulated by Ramkali at the end of the 12 weeks and whether she is able to send her daughter to school after 12 weeks or not. For this, we will use the concept of the sum of the series and check if the sum of the series is greater than Rs.2500 or not.

Complete step-by-step answer:
According to the question, Ramkali started saving with Rs.100 in the first week and gradually increased her savings by Rs.20 every week so as to achieve the target of Rs.2500 at the end of 12 weeks. So, we need to determine the sum of the total savings.
Mathematically, we can write
$
\Rightarrow Saving = 100 + (100 + 20) + (120 + 20) + {...._{12{\text{ weeks}}}} \\
   = 100 + 120 + 140 + {...._{12{\text{ weeks}}}} \;
 $
The sum of the arithmetic series is given as:
$S = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]$ where, ‘n’ is the total number of terms in the series, ‘a’ is the first term of the series and ‘d’ is the common difference between the terms of the series.
Here, the total number of terms in the series is 12 with the first term 100 and the common difference 20.
Substituting the values in the formula
$S = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]$ to determine the total saving by Ramkali.
$
\Rightarrow S = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right] \\
   = \dfrac{{12}}{2}\left[ {2 \times 100 + \left( {12 - 1} \right)20} \right] \\
   = 6\left[ {200 + 11 \times 20} \right] \\
   = 6\left[ {200 + 220} \right] \\
   = 6 \times 420 \\
   = 2520 \;
 $
So, the total saving by Ramkali is Rs.2520 which is obviously greater than Rs.2500 and hence, she will be able to send her daughter to school.
So, the correct answer is “2520”.

Note: It is very important to note here that the savings is increasing every week by Rs.20 with respect to previous weeks saved amount. Students often confuse the simultaneous increase in the amount with the fixed increase in the amount. Moreover, here as the difference between the terms is the same, we have used the formula for the arithmetic progression only.