Question

The sum of all two digit numbers which when divided by 4 leaves remainder 1 is

Answer 1

We need to find the sum of two digit numbers of the form 4k +1. We can see that these numbers form an arithmetic sequence with a common difference d. So to find the sum we need to find the number of terms using the formula ${a_n} = a + (n - 1)d$ and then find the sum using the formula ${S_n} = \frac{n}{2}(a + {a_n})$

Complete step-by-step answer:
We are asked for sum of two digit numbers which leaves a remainder 1 when divided by 4
Hence the numbers must be a successor of numbers which are multiple of four
We need to find the sum of two digit numbers of the form 4k + 1 , where
Step 2 :
So from this we get the numbers to be 13 , 17 , 21 , …… , 97
We can see that there exist a common between the numbers
(i.e.) 17 – 13 =4
21 – 17 =4
Therefore the numbers form an A.P with a common difference , d = 4
Step 3 :
In order to find the sum of these numbers we need to find the number of terms in this sequence.
Here , we have the first term to be 13 and the last term to be 97
(i . e.)$a = 13{\text{ and }}{a_n} = 97$
Since ${\text{ }}{a_n} = 97$, we know that the formula to find the nth term of the sequence is
$\Rightarrow {a_n} = a + (n - 1)d$
Lets substitute the known values in the above formula
$\Rightarrow 97 = 13 + (n - 1)4 \\\ \Rightarrow 97 - 13 = 4n - 4 \\\ \Rightarrow 84 = 4n - 4 \\\ \Rightarrow 84 + 4 = 4n \\\ \Rightarrow 88 = 4n \\\ \Rightarrow \dfrac{{88}}{4} = n \\\ \Rightarrow n = 22 \\\$
Hence there are 22 terms in the sequence.
Step 4 :
Now let's find the sum of the terms of the sequence.
Sum of the terms is given by the formula
$\Rightarrow {S_n} = \dfrac{n}{2}(a + {a_n})$
We have n = 22 and a = 13 and ${a_n} = 97$
Lets substitute it in the above formula
$\Rightarrow {S_{22}} = \dfrac{{22}}{2}(13 + 97) \\\ \Rightarrow {S_{22}} = 11(110) \\\ \Rightarrow {S_{22}} = 1210 \\\$

Therefore the sum of two digit numbers which when divided by 4 leaves a remainder 1 is 1210.

Note: The above formula for the sum of n terms can be used only if we know the last term of the sequence or series , when we do not know the last term we can use this alternate formula
$\Rightarrow {S_n} = \dfrac{n}{2}(2a + (n - 1)d)$