I had posted 2 blogs earlier on the subject-the first for familiarising the topic-arithmetical progressions and second for finding subsequent terms/sums of terms etc.
Let me now show you an interesting observation-
Consider the 10 terms in the following progression-
1,5,9,13,17,21,25,29,33,37
Step 1-I will now multiply the first with the last,the second with the penultimate and so on and tabulate the results-
1*37=37..,5*33=165...9*29=261...13*25=325...17*21=357 viz 37,165,261,325,357
Step2- I start finding the differences between successive values above starting from the last 357
357-325=32,...,325-261=64....261-165=96...165-37=128 viz 32,64,96,128
Step3-Again I find the differences.I will get all values equal to 32.
Similar procedure has been adopted for the 8 terms in another progression 1,4,7,10,13,16,19,22
to get the following results
Step1-22,76,112,130
Step 2-18,36,54
Step 3- All values equal to 18.
You can get similar results for any arithmetical progression,starting with any number,involving any common difference,and any number of terms-only the number of terms should be even and not odd,as in that case there will be a middle term which will have no companion for multiplication
You can predict what the final result in the final step will be in any such case.
For instance consider the progression starting with 7,having common difference 5 and a total of 32 terms.The final result at the final step will be-You will get all values equal to 50.This is 2 times the square of the common difference 5 viz 2*5*5=50,like in the earlier examples shown viz 2*4*4=32 and 2*3*3=18
Let me now show you an interesting observation-
Consider the 10 terms in the following progression-
1,5,9,13,17,21,25,29,33,37
Step 1-I will now multiply the first with the last,the second with the penultimate and so on and tabulate the results-
1*37=37..,5*33=165...9*29=261...13*25=325...17*21=357 viz 37,165,261,325,357
Step2- I start finding the differences between successive values above starting from the last 357
357-325=32,...,325-261=64....261-165=96...165-37=128 viz 32,64,96,128
Step3-Again I find the differences.I will get all values equal to 32.
Similar procedure has been adopted for the 8 terms in another progression 1,4,7,10,13,16,19,22
to get the following results
Step1-22,76,112,130
Step 2-18,36,54
Step 3- All values equal to 18.
You can get similar results for any arithmetical progression,starting with any number,involving any common difference,and any number of terms-only the number of terms should be even and not odd,as in that case there will be a middle term which will have no companion for multiplication
You can predict what the final result in the final step will be in any such case.
For instance consider the progression starting with 7,having common difference 5 and a total of 32 terms.The final result at the final step will be-You will get all values equal to 50.This is 2 times the square of the common difference 5 viz 2*5*5=50,like in the earlier examples shown viz 2*4*4=32 and 2*3*3=18
No comments:
Post a Comment