3. Fin the sum of 'n' terms of
the AP: n + (n+2) + (n+4) +
(n+6) ........
Answers
Answer:
Sum of N Terms Formula
The sum of n terms of AP is the sum(addition) of first n terms of the arithmetic sequence. It is equal to n divided by 2 times the sum of twice the first term – ‘a’ and the product of the difference between second and first term-‘d’ also known as common difference, and (n-1), where n is numbers of terms to be added.
Sum of n terms of AP = n/2[2a + (n – 1)d]
For example:
1, 4, 9, 16, 25, 36, 49 ……….625 represents a sequence of squares of natural numbers till 25.
3, 7, 11, 15, 19,………..87 forms another sequence, where each of the terms exceeds the preceding term by 4.
If all the terms of a progression except the first one exceeds the preceding term by a fixed number, then the progression is called arithmetic progression. If a is the first term of a finite AP and d is a common difference, then AP is written as – a, a+d, a+2d, ………, a+(n-1)d.
Note: Before learning how to derive a formula to get the sum of n terms in an AP, try this activity:
Try to get the sum of the first 100 natural numbers without using any formula.
This question was posed in the same way to one of the great mathematicians, Carl Gauss (1777-1855). He is often referred to be ‘Princeps mathematicorum’ (Latin), meaning ‘the foremost of mathematicians’. At that time, his age was 10 yrs. He came up with the answer to the above problem in a matter of seconds.
The proof for the question can be done using the following way:
The sum of the number can be represented as
Sum = 1+2+3+4+……………+ 97 + 98 + 99 + 100——————————————– (1)
Even if the order of the numbers is reversed, their sum remains the same.
Sum = 100 + 99 + 98 + 97 + ………..+ 4 + 3 + 2 + 1—————————————– (2)
Adding equations 1 and 2, we get
2 × Sum = (100+ 1) + (99+2) + (98+3 )+ (97 +4)+ ………..(4+97)+(3+98)+(2+99)+(1+100)
2 × Sum = 101 + 101 + 101 + 101 + ………..(4+97)+(3+98)+(2+99)+(1+100)
2 ×Sum = 101 (1 + 1 + 1 + …..100 terms)
2 × Sum = 101 (100)
Sum = {101 × 100}/{2}
Sum = 5050
Using the above method, sum of numbers like 1000, 10000, etc. can also be calculated.