MSC #20: Arithmetic Sequence – Counting terms
If you were asked how many numbers are there from 26 to 89, would you try to recall the formula taught to us in grade school or would you actually count it?
Today’s Math Special for Christmas deals with an easy way of counting the numbers of terms in a sequence. Counting, in Vedic Mathematics, is covered by the sutra “By One More than the One Before”, which will be discussed by MATH-inic founder Virgilio Y. Prudente in the opening session of the Inspirational Math from India (Year 4) on Dec 4 while “Linear sequences and summation of series.” will be the topic of IAVM chair James Glover’s talk on Dec 11, 2021
You can use this link to register for the event: https://bit.ly/IMI4Reg
Now, if you were asked how many numbers there are from 1 to 25, you KNOW that the answer is 25, because we always start to count from 1 and the last number is the total count.
Since we also KNOW that there are 89 numbers from 1 to 89, removing the first 25 will give us the count from 26 to 89, which is 89 – 25 = 64. In effect we transformed counting from 26 to 89 into counting from 1 to 64.
I Googled “how to find the number of terms in an arithmetic sequence” and I found this in a website: “Use the formula tn = a + (n – 1) d to solve for n. Plug in the last term (tn), the first term (a), and the common difference (d). Work through the equation until you’ve solved for n.”
A rearrangement of this formula is easier to use: n = (tn – a)/d + 1.
But the key to learning Math is in understanding not memorizing formulas. So, why bother to remember formulas when you can just transform the arithmetic sequence into consecutive counting numbers starting from 1 and immediately know the answer?
In our example, I purposely used “easy-to-compute” numbers so that we can focus on the method not on the calculations: “Find the number of terms in the sequence – 26, – 19, – 12…590.”
It is easy to notice that the common difference between the terms is 7. So we will first use by Addition or by Subtraction to convert the first term into the common difference. 7 – (-26) = 33. Thus we have to add 33 to – 26 to make it 7.
Next, we also add 33 to the last term, 590, to make it 623.
We, then divide the “new” first and last terms by the common difference 7, getting a new sequence of counting numbers from 1 to 89.
