Question

Two AP's have the same common difference. The difference between their I. \[100th\] term is \[100\] , what is the difference between their \[1000th\] terms?
II.How many three-digit numbers are divisible by \[7\] ?

Answer 1

Hint: For the first question, we will form an equation and find out the \[100th\] term of both the AP with the help of formula \[{t_n} = a + (n - 1)d\] . Then we will find their difference and substitute it in their \[1000th\] term to arrive at the answer.
For the second question, we will form an AP of three-digit numbers divisible by \[7\] and then find out the number of terms \[n\] using the formula, \[{t_n} = a + (n - 1)d\] .

Complete step by step solution:
I.Let the first AP be denoted by \[x\] and the second AP be denoted by \[y\] .
Their \[100th\] term using the formula \[{t_n} = a + (n - 1)d\] will be:
 \[{x_{100}} = x + (100 - 1)d = x + 99d\]
 \[{y_{100}} = y + (100 - 1)d = y + 99d\]
Now the difference between them is \[100\] that is-
 \[{x_{100}} - {y_{100}} = 100\]
Substituting the values, we get,
 \[x + 99d - (y + 99d) = 100\]
Opening the brackets, we get,
 \[x + 99d - y - 99d = 100\]
 \[x - y = 100\]
Now the difference between their \[1000th\] term can be found out as follows:
 \[{x_{1000}} - {y_{1000}} = [x + (1000 - 1)d] - [y + (1000 - 1)d] \]
 \[{x_{1000}} - {y_{1000}} = x + 999d - (y + 999d)\]
Opening the brackets, we get,
 \[{x_{1000}} - {y_{1000}} = x + 999d - y - 999d\]
 \[{x_{1000}} - {y_{1000}} = x - y\]
Substituting the value of \[x - y = 100\] , we get,
 \[{x_{1000}} - {y_{1000}} = 100\]
Thus, the difference between their \[1000th\] terms is \[100\] .

II.We will have to write an AP of three-digit numbers divisible by \[7\] . It will be \[105,112,119,....,994\]
Here, \[a = 105\] , \[d = 7\] and \[{t_n} = 994\] .
Now we can apply the formula and get:
 \[{t_n} = a + (n - 1)d\]
 \[994 = 105 + (n - 1)(7)\]
Solving the bracket, we get,
 \[994 = 105 + 7n - 7\]
Simplifying the equation, we get,
 \[7n = 994 - 105 + 7\]
 \[7n = 896\]
Dividing, we get,
 \[n = \dfrac{{896}}{7}\]
 \[n = 128\]
Hence, there are \[128\] three-digit numbers which are divisible by \[7\] .

Note: Arithmetic Progression (AP) is a sequence of numbers in order in which the difference of any two consecutive numbers is a constant value.
In the second question, we can use the following trick to find the numbers and construct an AP:
To check if a number is evenly divisible by \[7\] :
Take the last digit of the number and double it. For e.g. In \[994\] taking last digit- \[(4 \times 2 = 8)\]
Then subtract the result from the rest of the number i.e. \[99 - 8 = 91\]
If the resulting number is evenly divisible by \[7\] , so is the original number. i.e. \[\dfrac{{91}}{7} = 13\] so \[994\] is divisible by \[7\] .