Question

A man arranges to pay off a debt of RS. \[{\text{36}}00\]by \[{\text{4}}0\] annual installments which form an A.P. When \[{\text{3}}0\] of the installments are paid, he dies leaving one - third of the debt unpaid, find the value of the first installment.

Answer 1

Hint:A sequence is a list of items or objects which have been arranged in a sequential way.
A series can be highly generalized as the sum of all the terms in a sequence. However, there has to be a definite relationship between all the terms of the sequence.
Arithmetic sequence
A sequence in which every term is created by adding or subtracting a definite number to the preceding number is an arithmetic sequence.
                              A.P
Sequence \[{\text{a}},{\text{ a }} + {\text{ d}},{\text{ a }} + {\text{ 2d}},{\text{ }}. \ldots \ldots \ldots \ldots \ldots {\text{a }} + {\text{ }}\left( {{\text{n }}-{\text{ 1}}} \right){\text{d}}\]
General term (nth term) \[{{\text{a}}_{\text{n}}} = {\text{ a }} + {\text{ }}\left( {{\text{n }}-{\text{ 1}}} \right){\text{d}}\]
Therefore,
The value of the first installment is\[{\text{51}}\].

Complete step by step answer:
Given:
Total amount of Debt, \[\]
Number of annual installments, \[n = {\text{ 4}}0\]
He paid \[{\text{3}}0\] installment and he die leaving \[\dfrac{1}{3}\] of the debt unpaid.
Unpaid amount = \[\dfrac{1}{3}{\text{ }} \times {\text{ }}3600{\text{ }} = {\text{ }}1200\]
Total payment he paid in \[{\text{3}}0\] installment,\[\]
\[{{\text{S}}_{{\text{3}}0}}{\text{ }} = {\text{ 24}}00\]
By using the formula, Sum of nth terms, \[{\text{Sn }} = {\text{ }}\dfrac{{\text{n}}}{2}{\text{ }}\left[ {{\text{2a }} + {\text{ }}\left( {{\text{n }}-{\text{ 1}}} \right){\text{ d}}} \right]\]
For \[{\text{3}}0\] installments:
\[{{\text{S}}_{{\text{3}}0}}{\text{ }} = {\text{ }}\dfrac{{{\text{3}}0}}{2}{\text{ }}\left[ {{\text{2a }} + {\text{ }}\left( {{\text{3}}0{\text{ }} - {\text{ 1}}} \right){\text{d}}} \right]\]

\[{\text{24}}00{\text{ }} = {\text{ 15 }}\left[ {{\text{2a }} + {\text{ 29d}}} \right]\]
\[\dfrac{{{\text{24}}00}}{{15}}{\text{ }} = {\text{ }}\;\left[ {{\text{2a }} + {\text{ 29d}}} \right]\]
\[{\text{16}}0{\text{ }} = {\text{ 2a }} + {\text{ 29d}}\]
\[{\text{2a }} = {\text{ 16}}0{\text{ }} - {\text{ 29d}}\]
\[{\text{2a }} + {\text{ 29d }} = {\text{ 16}}0{\text{ }} \ldots \ldots ..\left( {\text{1}} \right)\]
For \[{\text{4}}0\] installments:
\[{{\text{S}}_{{\text{4}}0}}{\text{ }} = {\text{ }}\dfrac{{{\text{4}}0}}{2}{\text{ }}\left[ {{\text{2a }} + {\text{ }}\left( {{\text{n }} - {\text{ 1}}} \right){\text{ d}}} \right]\]
\[{\text{36}}00{\text{ }} = {\text{ 2}}0{\text{ }}\left[ {{\text{2a }} + {\text{ }}\left( {{\text{4}}0{\text{ }} - {\text{1}}} \right){\text{ d}}} \right]\]
\[\dfrac{{{\text{36}}00}}{2}{\text{ }} = {\text{ 2a }} + {\text{ 39d}}\]
\[{\text{18}}0{\text{ }} = {\text{ 2a }} + {\text{ 39d}}\]
\[{\text{2a }} + {\text{ 39d }}\; = {\text{ 18}}0{\text{ }} \ldots \ldots \ldots ..\left( {\text{2}} \right)\]
On subtracting eq (i) from (ii), we get
\[{\text{2a }} + {\text{ 39d }}\; = {\text{ 18}}0\]
\[{\text{2a }} + {\text{ 29d }} = {\text{ 16}}0\]
\[\;\left( - \right){\text{ }}\;\left( - \right){\text{ }}\;{\text{ }}\;{\text{ }}\;\left( - \right)\]
-----------
  \[{\text{1}}0{\text{d }} = {\text{ 2}}0\]
\[{\text{d }} = {\text{ }}\dfrac{{{\text{2}}0}}{{10}}\]

\[{\text{d }} = {\text{ 2}}\]
On Putting the value of \[d{\text{ }} = {\text{ }}2{\text{ }}in{\text{ }}eq{\text{ }}\left( 1 \right),\]
\[{\text{2a }} + {\text{ 29d }} = {\text{ 16}}0\]
\[{\text{2a }} + {\text{ 29 }}\left( {\text{2}} \right){\text{ }} = {\text{ 16}}0\]
\[{\text{2a }} + {\text{ 58 }} = {\text{ 16}}0\]
\[{\text{2a }} = {\text{ 16}}0{\text{ }} - {\text{ 58}}\]
\[{\text{2a }} = {\text{ 1}}0{\text{2}}\]
\[{\text{a }} = \dfrac{{{\text{ 1}}0{\text{2}}}}{2}\]
\[{\text{a }} = {\text{ 51}}\]
Hence, the value of the first installment is \[{\text{Rs51}}.\]

Note:
The series is finite or infinite depending on the situation whether the sequence is finite or infinite. Finite sequences and series have defined first and last, terms, whereas infinite sequences and series continue indefinitely.