Question

Ramkali saved Rs.5 in the first week of a year and then increased her weekly saving by Rs.1.75. If in the \[{{n}^{th}}\] week her weekly savings become Rs.20.75, find n.
(a) n = 10
(b) n = 12
(c) n = 16
(d) n = 20

Answer 1

Hint: Find the savings of Ramkali in \[{{1}^{st}}\] week, \[{{2}^{nd}}\] week, \[{{3}^{rd}}\] week. Comparing these values they formulate into series. The \[{{n}^{th}}\] week savings will be her last term. Substitute these values and find n i.e. find the number of weeks.

Complete step-by-step answer:

The savings made by Ramkali in the first week = Rs.5.

Savings made in the \[{{2}^{nd}}\] week = 5 + 1.75 = 6.75.

Thus the savings made by her in the \[{{3}^{rd}}\] week = 5 + 1.75 + 1.75 = Rs.8.50.

Thus it forms a series 5, 6.75, 805,……

As the differences between the terms are the same, it's an arithmetic progression.

So the common difference is denoted by d.

\[\therefore d=1.5\]

The first term, a = 5.

It is given that in \[{{n}^{th}}\] week, her savings is Rs.20.75.

Thus the last term of the series becomes 20.75.

i.e. 5, 6.75, 8.50,……, 20.75

Let us denote the last – term as l.

\[\therefore l=20.75\]

We need to find the value of n i.e. we need to find the number of weeks.

We know the formula in AP,

\[{{a}_{n}}=a+\left( n-1 \right)d\]

\[{{a}_{n}}=l=20.75\], a = 5, d = 1.5.

Substitute the values in the equation and find n.

\[\begin{align}
  & 20.75=5+\left( n-1 \right)\times 1.5 \\
 & 20.75-5=\left( n-1 \right)1.5 \\
 & \therefore n-1=\dfrac{20.75-5}{1.5} \\
 & \therefore n-1=\dfrac{15.75}{1.5} \\
 & \therefore n-1=9 \\
 & \therefore n=10 \\
\end{align}\]

Hence in the \[{{10}^{th}}\] week, her savings will become Rs.20.75.

\[\therefore \] Option (a) is the correct answer.

Note: Without series for every check you can also add up the value.

\[{{1}^{st}}\] = 5

\[{{2}^{nd}}\] = 5 + 1.75 = 6.75

\[{{3}^{rd}}\] = 6.75 + 1.75 = 8.50

\[{{4}^{th}}\] = 8.50 + 1.75 = 10.25

\[{{5}^{th}}\] = 10.25 + 1.75 = 12

\[{{6}^{th}}\] = 12 + 1.75 = 13.75

\[{{7}^{th}}\] = 13.75 + 1.75 = 15.5

\[{{8}^{th}}\] = 15.5 + 1.75 = 17.25

\[{{9}^{th}}\] = 17.25 + 1.75 = 19

\[{{10}^{th}}\] = 19 + 1.75 = 20.75

Thus in \[{{10}^{th}}\] week we got Rs.20.75.