# A man arranges to pay off a debt of pounds $3600$ by $40$ annual installments which form an arithmetic series. When $30$ of the installments are paid he dies leaving a third of the debt unpaid. find the value of the first installment.

Answer

Verified

327.9k+ views

Hint-Use sum of n terms of an A.P. As the given conditions form an A.P.

According to the question, the total amount of debt to be paid in \[40\] installments is $3600$ pounds

So, the sum of all $40$ installments ${S_{40}} = 3600$

After $30$installments one-third of his debt is left unpaid this means that he paid two third of the payment.

So, the amount he paid in $30$ installments

$

= \frac{2}{3} \times 3600 = 2400 \\

\\

$

Sum of first $30$ installments is ${S_{30}} = 2400$

Let us take the first installments as $a$ and common difference as $d$.

So, using the formula for the sum of $n$ terms of an A.P.

${S_n} = \frac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$

Let us find $a$ and$d$ for $30$ installments

$

{S_{30}} = \frac{{30}}{2}\left[ {2a + \left( {30 - 1} \right)d} \right] = 2400 \\

\Rightarrow 15\left( {2a + 29d} \right) = 2400 \\

\Rightarrow 2a + 29d = 160...........\left( 1 \right) \\

$

As we know the sum of all $40$ installments is ${S_{40}} = 3600$

$

{S_{40}} = \frac{{40}}{2}\left[ {2a + \left( {40 - 1} \right)d} \right] = 3600 \\

\Rightarrow 20\left( {2a + 39d} \right) = 3600 \\

\Rightarrow 2a + 39d = 180...........\left( 2 \right) \\

$

Solve for (1) and (2) equations

$

\Rightarrow 2a + 30d - 2a - 29d = 180 - 160 \\

\Rightarrow 10d = 20 \\

\Rightarrow d = 2 \\

$

Put value of $d$ in any equations

$

\Rightarrow 2a + 29 \times 2 = 160 \\

\Rightarrow 2a + 58 = 160 \\

\Rightarrow 2a = 102 \\

\Rightarrow a = 51 \\

$

So, the first installment is 51 pounds.

Note- Whenever we come to these types of problems first of all find the sum of all installments and also find the sum of rest of installments then after using sum of $n$ terms of an AP So, we can easily calculate the first installment.

According to the question, the total amount of debt to be paid in \[40\] installments is $3600$ pounds

So, the sum of all $40$ installments ${S_{40}} = 3600$

After $30$installments one-third of his debt is left unpaid this means that he paid two third of the payment.

So, the amount he paid in $30$ installments

$

= \frac{2}{3} \times 3600 = 2400 \\

\\

$

Sum of first $30$ installments is ${S_{30}} = 2400$

Let us take the first installments as $a$ and common difference as $d$.

So, using the formula for the sum of $n$ terms of an A.P.

${S_n} = \frac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$

Let us find $a$ and$d$ for $30$ installments

$

{S_{30}} = \frac{{30}}{2}\left[ {2a + \left( {30 - 1} \right)d} \right] = 2400 \\

\Rightarrow 15\left( {2a + 29d} \right) = 2400 \\

\Rightarrow 2a + 29d = 160...........\left( 1 \right) \\

$

As we know the sum of all $40$ installments is ${S_{40}} = 3600$

$

{S_{40}} = \frac{{40}}{2}\left[ {2a + \left( {40 - 1} \right)d} \right] = 3600 \\

\Rightarrow 20\left( {2a + 39d} \right) = 3600 \\

\Rightarrow 2a + 39d = 180...........\left( 2 \right) \\

$

Solve for (1) and (2) equations

$

\Rightarrow 2a + 30d - 2a - 29d = 180 - 160 \\

\Rightarrow 10d = 20 \\

\Rightarrow d = 2 \\

$

Put value of $d$ in any equations

$

\Rightarrow 2a + 29 \times 2 = 160 \\

\Rightarrow 2a + 58 = 160 \\

\Rightarrow 2a = 102 \\

\Rightarrow a = 51 \\

$

So, the first installment is 51 pounds.

Note- Whenever we come to these types of problems first of all find the sum of all installments and also find the sum of rest of installments then after using sum of $n$ terms of an AP So, we can easily calculate the first installment.

Last updated date: 24th May 2023

â€¢

Total views: 327.9k

â€¢

Views today: 4.86k

Recently Updated Pages

If a spring has a period T and is cut into the n equal class 11 physics CBSE

A planet moves around the sun in nearly circular orbit class 11 physics CBSE

In any triangle AB2 BC4 CA3 and D is the midpoint of class 11 maths JEE_Main

In a Delta ABC 2asin dfracAB+C2 is equal to IIT Screening class 11 maths JEE_Main

If in aDelta ABCangle A 45circ angle C 60circ then class 11 maths JEE_Main

If in a triangle rmABC side a sqrt 3 + 1rmcm and angle class 11 maths JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE