Question

A child puts one five rupee coin her saving in the piggy bank on the first day. She increases her saving by one five rupee-coin daily. If the piggy bank can hold $190$ coins of five rupees in all, find the number of days she continue to put the five rupee coins into it and find the total money she saved.

Answer 1

$\to$ Child's daily input is 5, 10, 15, ---

& the number of coins on each days are

1, 2, 3, _ _ _ $\to$ A.P with a = 1, d = 1 where $S_n⩽190$

$\Rightarrow \frac{n}{2}(2+(n-1\left)1\right)⩽190\Rightarrow n+\frac{n^2}{2}-\frac{n}{2}⩽190$

$\Rightarrow n^2+n⩽380\Rightarrow n^2+n-380⩽0\Rightarrow n^2+20n-19n-380⩽0$

$\Rightarrow n(n+20)-19(n+20)⩽0\Rightarrow (n-19)(n+20)⩽0$

$\Rightarrow nϵ[-20,19]\Rightarrow$ she can

put coins for 19 days

For amount, $5,10,15,...\to AP=5$ (1, 2, 3,_ _ _ )

Sum of 19 terms $=\frac{5\times 19\left(20\right)}{2}=Rs.950$