Question

Find the number of two digit multiples of 4, and the sum of them.

Answer 1

Hint – In this particular type of question use the concept that the last term of an A.P is given as ${a_n} = a + \left( {n - 1} \right)d$, and the sum of A.P series is given as ${S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$, where symbols have their usual meanings so use these concepts to reach the solution of the question.

Complete step by step solution:
As we all know that the first two digit number is 10 and the last two digit number is 99.
But none of these is divided by 4.
The first two digit number which is a multiple of 4 is 12 and second two digit number which is the multiple of 4 is 16 and the last two digit number which is the multiple of 4 is 96.
So the set of two digits numbers which are multiple of 4 is = {12, 16, ..............., 96}.
Now as we see that the above series form an arithmetic progression (A.P), whose first term (a) is 12, common difference (d) = (16 – 12) = (20 – 16) = 4 and last term (${a_n}$) = 96.
Now as we know that in an A.P the last term (${a_n}$) formula is given as
$\Rightarrow {a_n} = a + \left( {n - 1} \right)d$, where n is the number of terms.
Now substitute the values we have,
$\Rightarrow 96 = 12 + \left( {n - 1} \right)4$
Now simplify this we have,
$\Rightarrow 96 - 12 = 84 = 4\left( {n - 1} \right)$
$\Rightarrow \dfrac{{84}}{4} = 21 = \left( {n - 1} \right)$
$\Rightarrow n = 21 + 1 = 22$
So there are 22 two digit terms in the series which is multiple of 4.
Now we have to find out the sum of these 22 terms.
Now as we know that the sum (${S_n}$) of the A.P series is given as,
$\Rightarrow {S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$, where symbols have their usual meanings.
Now substitute the values we have,
$\Rightarrow {S_n} = \dfrac{{22}}{2}\left( {2\left( {12} \right) + \left( {22 - 1} \right)4} \right)$
Now simplify we have,
$\Rightarrow {S_n} = 11\left( {24 + 84} \right) = 11\left( {108} \right) = 1188$
So this is the required answer.

Note – Whenever we face such types of question the key concept we have to remember is always recall all the formulas of an Arithmetic series which is all stated above then first find out the number of terms in the series as above, then find out the sum of the given series and simplify as above, we will get the required answer.