Question

i) Study the pattern
\[\begin{array}{l}
91 \times 11 \times 1 = 1001\\
91 \times 11 \times 2 = 2002\\
91 \times 11 \times 3 = 3003
\end{array}\]
Write the next seven steps.Check weather the result is correct.
ii) Try the pattern for
\[\begin{array}{l}
143 \times 7 \times 1\\
143 \times 7 \times 2\\
143 \times 7 \times 3……\\
\end{array}\]

Answer 1

Hint: Check the last number with which we are multiplying in both (i) and (ii) and then check the answer given on the right hand side the answer is changing with the change of last number only as the rest of the number remains the same throughout. So, there's a relation of the last number with the answer. Try to use that in both the questions.

Complete step-by-step answer:
For question (i) it has been observed that the last number with which we are multiplying \[91 \times 11\] is ultimately changing the answer and it is directly proportional, because the number we are multiplying with is the first and last digit of the answer and the remaining two digits in the middle will always be zero i.e., for \[91 \times 11 \times 2\] will be 2002 thus the two digits in the middle is always zero and the first and last digit is same as the number we are multiplying lastly.But this trend will not be held for the last number i.e., 10 because we know if we multiply any number by 10 it will just increase 0 in last then what we have got by multiplying with 1. Therefore, instead of 100010 we will get 10010. So by using this property of the pattern the next steps after \[91 \times 11 \times 3 = 3003\] will be
\[\begin{array}{l}
91 \times 11 \times 4 = 4004\\
91 \times 11 \times 5 = 5005\\
91 \times 11 \times 6 = 6006\\
91 \times 11 \times 7 = 7007\\
91 \times 11 \times 8 = 8008\\
91 \times 11 \times 9 = 9009\\
91 \times 11 \times 10 = 10010
\end{array}\]

You can use a calculator or just multiply manually to verify the result.
For question (ii) the same trend will be followed as the last number increases and the ans will change because the rest of the numbers remain the same. First of all let us try to find out \[143 \times 7 \times 1\] the answer will be 1001. So just like the last pattern this trend will remain the same as a result we will get the pattern as
\[\begin{array}{l}
143 \times 7 \times 1 = 1001\\
143 \times 7 \times 2 = 2002\\
143 \times 7 \times 3 = 3003\\
143 \times 7 \times 4 = 4004\\
143 \times 7 \times 5 = 5005\\
143 \times 7 \times 6 = 6006\\
143 \times 7 \times 7 = 7007\\
143 \times 7 \times 8 = 8008\\
143 \times 7 \times 9 = 9009\\
143 \times 7 \times 10 = 10010\\
.\\
.\\
.\\
.\\

\end{array}\]
We can again just multiply all the terms manually or do it using a calculator to verify the results.


Note: Observing the pattern in which the numbers were changing are really necessary or it will take a lot of time calculating each one of them manually. An alternative approach to the problem would be that if you can only multiply the first step and observe that as the last number we are multiplying is basically forming a table of the first term and it is really easy to predict a table if the number only consists of 0’s and 1’s because the 0’s will remain the same and the 1’s will change according to the last number we are multiplying.