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.

& 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{aligned}$$ Hence in the $${{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.