# Find the sum of all positive integers, from $5$ to $1555$ inclusive, that are divisible by $5$

A.$242489$

B.$242580$

C.$242420$

D.$252420$

Answer

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Hint : (Use the given series to find first term and common difference and proceed as considering the series an AP)

The first few terms of a sequence of positive integers divisible by $5$ is given by

\[5,10,15,...\]

The above sequence has a first term ${a_1} = 5$ and a common difference \[d = 5\].

We need to know the rank of the term $1555$.

We use the formula for the ${n^{th}}$ term as follows :- ${a_n} = a + (n - 1)d$

Applying the formula

\[1555 = {a_1} + (n - 1)d\]

Substituting the values of ${a_{1\,}}$ and $d$ we get,

\[1555 = 5 + 5(n - 1)\]

Solve to obtain $n$

\[n = 311\]

We now know that $1555$ is the ${311^{th}}$ term, we can use the formula for the sum in AP .

We Know

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

We have to calculate ${S_{311}}$

$

{S_{311}} = \dfrac{{311}}{2}\left( {(2)5 + 310(5)} \right) = 242580 \\

\\

$

Hence the Correct option is B.

Note :- In these types of questions we have to consider the given series as an AP, then solve it by using the formulas of nth term of AP and sum of N terms of an AP to get the result as done above.

The first few terms of a sequence of positive integers divisible by $5$ is given by

\[5,10,15,...\]

The above sequence has a first term ${a_1} = 5$ and a common difference \[d = 5\].

We need to know the rank of the term $1555$.

We use the formula for the ${n^{th}}$ term as follows :- ${a_n} = a + (n - 1)d$

Applying the formula

\[1555 = {a_1} + (n - 1)d\]

Substituting the values of ${a_{1\,}}$ and $d$ we get,

\[1555 = 5 + 5(n - 1)\]

Solve to obtain $n$

\[n = 311\]

We now know that $1555$ is the ${311^{th}}$ term, we can use the formula for the sum in AP .

We Know

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

We have to calculate ${S_{311}}$

$

{S_{311}} = \dfrac{{311}}{2}\left( {(2)5 + 310(5)} \right) = 242580 \\

\\

$

Hence the Correct option is B.

Note :- In these types of questions we have to consider the given series as an AP, then solve it by using the formulas of nth term of AP and sum of N terms of an AP to get the result as done above.

Last updated date: 23rd Sep 2023

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