Question

Calculate the sum of even numbers between $12\;$ and $90\;$ which are divisible by $8$
A. $500\;$
B. $510\;$
C. $520\;$
D. $620\;$

Answer 1

In questions like these start first with finding the first number and last between the given integers and find the sequence needed. The sum needed can be found using$Sum=\dfrac{n}{2}\left( a+{{a}_{n}} \right)$. We don’t know the number of integers between the first and last numbers in this arithmetic sequence that’s why we find it using ${{a}_{n}}=a+\left( n-1 \right)d$

Formulas used:
${{a}_{n}}=a+\left( n-1 \right)d$
$Sum=\dfrac{n}{2}\left( a+{{a}_{n}} \right)$

Complete step by step answer:
The first number of the arithmetic sequence is $16\;$ because it’s the first number after $12\;$ which is divisible by $8$. The number after this will be $24\;$. Now the last number of this sequence is $88\;$ because it’s the first number before $90\;$ which is divisible by $8$. In this way the arithmetic sequence will be$16,24,32....88$.

Now, to find the number of integer in this sequence we use the formula ${{a}_{n}}=a+\left( n-1 \right)d$ where ${{a}_{n}}$ stands for last integer in the sequence a stands for the first integer in the sequence, d stands for the difference between two consecutive numbers in the sequence and n stands for the total number of numbers in the sequence. Therefore;
${{a}_{n}}=a+\left( n-1 \right)d$
Substituting the given values,
$88=16+(n-1)8$
Subtracting,
$72=(n-1)8$
Dividing both sides by $8$ we get,
$\dfrac{72}{8}=n-1$
Solving the fraction,
$9=n-1$
$\Rightarrow n=10$

Now through this we now know that there are $10\;$ integers in the sequence. Sum of an arithmetical equation is now found using the formula $Sum=\dfrac{n}{2}\left( a+{{a}_{n}} \right)$. We already know an ${{a}_{n}}$ and we found n using the first equation. So now substituting these values in the value for sum
$Sum=\dfrac{10}{2}\left( 16+88 \right)$
As we follow BODMAS to solve any equation we first solve the bracket
$Sum=\dfrac{10}{2}\left( 104 \right)$
Opening the bracket and multiplying
$Sum=\dfrac{10\times 104}{2}$
Dividing we get
$Sum=5\times 104$
$\therefore Sum=520$
Therefore we know the sum of even integers between $12\;$ and $90\;$ is $520$.

Hence, the correct answer is option C.

Note: A sequence is a set of things usually number that are in order. In an arithmetic equation the difference between two consecutive numbers is constant which is how we can derive the above used formula. It is noticed that since we need sum between numbers divisible by $8$ they form a pattern with that difference being $8$ which makes it known that the integers make an arithmetic sequence.