Question

Calculate the sum of numbers between $100{\text{ and }}150$ which are divisible by $13$.

A. $494$
B. $410$
C. $420$
D. $430$

Answer 1

Hint- In order to solve this problem first we will form an AP between $100{\text{ and }}150$ which are divisible by $13$, further we will find common difference and first term and at last by using the formula of sum of nth series we will get the required answer.

Complete step-by-step answer:

The numbers between $100{\text{ and }}150$ which are divisible by $13$are $104,117,.................143$

So AP = $104,117,.................143$$$2$$

Here first term is $104$

and common difference = second term $-$ first term $= {\text{ }}117 - 104{\text{ }} = {\text{ }}13$

As we know that

The sum of first $n$terms of arithmetic series formula can be written as

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

Where $n$= number of terms = ?
$a =$ first odd number $= 104$
$d =$common difference of A.P. $= 13$
${T_n} =$Last term $= 143$

So first we have to calculate total number of terms

As we know that ${n^{th}}$ term of AP is given as

${T_n} = a + (n - 1)d$

Substitute the values in above formula we have

$$143 = 104 + (n - 1)13 \\\ 143 = 104 + 13n - 13 \\\ 143 - 104 + 13 = 13n \\\ 52 = 13n \\\ n = 4 \\\$$

Therefore total number of terms $= 4$

Now put the values of $n,a,d$ and ${T_n}$ in formula of sum of $n$ term of series

$${S_4} = \dfrac{4}{2}\left[ {2 \times 104 + (4 - 1)13} \right] \\\ = 2[208 + (3)13] \\\ = 2[208 + 39] \\\ = 2[247] \\\ = 494 \\\$$

Hence, the sum of numbers between $100{\text{ and }}150$ which are divisible by $13$ is $494$ and the correct answer is option A.

Note- The formula states that the sum of our arithmetic sequence's first $n$ terms is equal to $n$ divided by $2$ times the sum of twice the beginning term, $a$, and the product of d, the common difference, and $n$ minus $1$. The n represents the number of words we put together.