Question

A man repays a loan of Rs.3250 by paying Rs.20 in the first month and then increase the
payment by Rs. 15 every month .How long will it take him to clear the loan?.

Plz solve
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Answer 1

$\bf{ Given} \begin{cases} \sf{\blue{ A \:or \:The Payment \:in\:first\:month\:= Rs.20 }} & \\\\ \sf{\pink{ D \:or\:Increase \:in\:Payment \:per\:month\:= \:Rs.15 }} & \\\\ \sf{\red{S_{n} \:or\:Total \:Amount \:of\:loan \:= \:Rs.3250}}\end{cases}$

Need To Find : n or How long will it take him to clear the loan .

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❍ Let's Consider the loan will be cleared in n months.

$\frak{\underline {\dag As, \:We \:know\:that,\:}}$

⠀⠀⠀⠀⠀ $\underline {\boxed { \sf{ \bigstar S_{n} = \dfrac{n}{2} \Bigg( 2a + ( n- 1)d \Bigg) }}}$

Where,

• a is the first term or First month's Payment in Rs. , d is the Difference or Increase in Payment every month & n is number of months to clear the loan . And we have already Given with $\bf{S_{n}}$ is Sum or the total Amount of Loan is Rs.3250 .

⠀⠀⠀⠀⠀⠀$\underline {\bf{\star\:Now \: By \: Substituting \: the \: Given \: Values \::}}$

⠀⠀⠀⠀⠀ $:\implies \tt{ Rs.3250 = \dfrac{n}{2} \Bigg( 2\times 20 + ( n- 1) 15 \Bigg) }$

⠀⠀⠀⠀⠀ $:\implies \tt{ Rs.3250 = \dfrac{n}{2} \Bigg( 40 + ( n- 1) 15 \Bigg) }$

⠀⠀⠀⠀⠀ $:\implies \tt{ 3250 \times 2 = n \Bigg( 40 + ( n- 1) 15 \Bigg) }$

⠀⠀⠀⠀⠀ $:\implies \tt{ 6500 = n \Bigg( 40 + ( n- 1) 15 \Bigg) }$

⠀⠀⠀⠀⠀ $:\implies \tt{ 6500 = n \Bigg( 40 + 15 n- 15 \Bigg) }$

⠀⠀⠀⠀⠀ $:\implies \tt{ 6500 = n \Bigg( 25 + 15 n \Bigg) }$

⠀⠀⠀⠀⠀ $:\implies \tt{ 6500 = 25n + 15 n^{2} }$

Or,

⠀⠀⠀⠀⠀ $:\implies \tt{ 15n^{2} + 25n - 6500 = 0 }$

$\sf{\bigstar Now \;Taking \:\bf{5} \: as \:common \:in \:each \: term -: }$

⠀⠀⠀⠀⠀ $:\implies \tt{ \cancel {15n^{2}} + \cancel {25n} - \cancel {6500} = 0 }$

⠀⠀⠀⠀⠀ $:\implies \tt{ 3n^{2} + 5n - 1300 = 0 }$

⠀⠀⠀⠀⠀ $:\implies \tt{ 3n^{2} - 60 n + 65n - 1300 = 0 }$

⠀⠀⠀⠀⠀ $:\implies \tt{ 3n(n - 20 ) 65( n - 20) = 0 }$

⠀⠀⠀⠀⠀ $:\implies \tt{ (n - 20 ) ( 3n - 65) = 0 }$

Then ,

⠀⠀⠀⠀⠀ $:\implies \tt{ (n - 20 ) = 0 }$

⠀⠀⠀⠀⠀ $:\implies \tt{ n = 20 }$

And ,

⠀⠀⠀⠀⠀ $:\implies \tt{ (3n + 65 ) = 0 }$

⠀⠀⠀⠀⠀ $:\implies \tt{ 3n = -65 }$

⠀⠀⠀⠀⠀ $:\implies \tt{ n = \dfrac{-65}{3} }$

Therefore,

⠀⠀⠀⠀⠀ $:\implies \tt{ n = \dfrac{-65}{3} \:or\:20 }$

⠀⠀⠀⠀⠀As We know that we are asked for Time and time cannot be in negative so by ignoring negative value;

⠀⠀⠀⠀⠀$\underline {\boxed {\pink{ \frak{ n = 20 }}}}\:\bf{\bigstar}$

Therefore,

⠀⠀⠀⠀⠀$\therefore \underline {\sf{ Hence ,\:The \:man \;will \:pay\:loan \:in\:\bf{20\:months}}}$

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Answer 2

Answer:

20,35,50...3250

an= a+(n-1)d

3250= 20+(n-1)15

3250-20= (n-1)15

3230/15=n

n=215.333=21 months