Introduction for Arithmetic Series function:
A sequence that is 3 + 7 + 11 + 15 + ··· + 99 or 10 + 20 + 30 + ··· + 1000 that includes an even distinction between terms. The first term known as a1, the general difference is d, and the number of terms is n. The calculation of an arithmetic sequence is established by multiplying the number of terms times the average of the 1st and last terms.
Proof of Arithmetic Series Function:
1) Addition of i as i moves from 1 to n: 1 + 2 + 3 + ... + n function.
2) Adding the calculation of adding the 2nd summation backwards:
1 + ... + n
n + ... + 1
3) The 1st equation as (i), and then the 2nd calculation becomes (n - i + 1)
4) Hence 2 * the equation is (i) + (n - i + 1) or (n + 1) for every item in the sequence.
5) Here there are n terms in the sequence, the 2 * calculation of value is n * (n + 1)
6) For a particular series, we have end up with (n * (n + 1) / 2)
Example Problem for Arithmetic Series Function:
Here we will take more time to assuming about the rule, but the formula for the calculation of a finite arithmetic series, frequently symbolized by sn is
Arithmetic series function Formula: Sn = n (`(a1 + an) / 2` ) or `n / 2` [2a1 + (n - 1) d]
Problem 1:
4 + 8 + 12 + 16 + ··· + 96 has a1 = 4 and d = 4. To find n, use the formula for an arithmetic sequence.
Solution:
We solve 4 + (n – 1) *4 = 96 to get n = 24.
Sum = 24 (4 +` 99 / 2` ) = 1236
Or
Sum = `24 / 2 ` [2* 4 + (24 – 1) *4] = 1236.
Problem 2:
How much will a bank staff receives over his 34 year career if his starting salary is 40,000, and he receives a 1,000 salary increments for every year he works here?
Solution
We have the sequence:
40,000 + 41,000 + 42,000 + ... + 73,000
= `34 / 2` (40,000 + 73,000) = 1,92,1000.
Problem 3:
Find S9 for the series an = 6n – 1
Solution:
The formula tells us we have to be known n, the 1st term, and the nth term n = 9 because we have to find out the sum of the first 9 terms.
a1 = 6(1) – 1 = 6 – 1 = 5
a 9 = 6 (9) - 1 = 54 – 1 = 53
Substitute these values into the equation gives
S9 = 9 (5 + `53 / 2)` = 9 (`58 / 2)` = 9 (29) = 261
A sequence that is 3 + 7 + 11 + 15 + ··· + 99 or 10 + 20 + 30 + ··· + 1000 that includes an even distinction between terms. The first term known as a1, the general difference is d, and the number of terms is n. The calculation of an arithmetic sequence is established by multiplying the number of terms times the average of the 1st and last terms.
Proof of Arithmetic Series Function:
1) Addition of i as i moves from 1 to n: 1 + 2 + 3 + ... + n function.
2) Adding the calculation of adding the 2nd summation backwards:
1 + ... + n
n + ... + 1
3) The 1st equation as (i), and then the 2nd calculation becomes (n - i + 1)
4) Hence 2 * the equation is (i) + (n - i + 1) or (n + 1) for every item in the sequence.
5) Here there are n terms in the sequence, the 2 * calculation of value is n * (n + 1)
6) For a particular series, we have end up with (n * (n + 1) / 2)
Example Problem for Arithmetic Series Function:
Here we will take more time to assuming about the rule, but the formula for the calculation of a finite arithmetic series, frequently symbolized by sn is
Arithmetic series function Formula: Sn = n (`(a1 + an) / 2` ) or `n / 2` [2a1 + (n - 1) d]
Problem 1:
4 + 8 + 12 + 16 + ··· + 96 has a1 = 4 and d = 4. To find n, use the formula for an arithmetic sequence.
Solution:
We solve 4 + (n – 1) *4 = 96 to get n = 24.
Sum = 24 (4 +` 99 / 2` ) = 1236
Or
Sum = `24 / 2 ` [2* 4 + (24 – 1) *4] = 1236.
Problem 2:
How much will a bank staff receives over his 34 year career if his starting salary is 40,000, and he receives a 1,000 salary increments for every year he works here?
Solution
We have the sequence:
40,000 + 41,000 + 42,000 + ... + 73,000
= `34 / 2` (40,000 + 73,000) = 1,92,1000.
Problem 3:
Find S9 for the series an = 6n – 1
Solution:
The formula tells us we have to be known n, the 1st term, and the nth term n = 9 because we have to find out the sum of the first 9 terms.
a1 = 6(1) – 1 = 6 – 1 = 5
a 9 = 6 (9) - 1 = 54 – 1 = 53
Substitute these values into the equation gives
S9 = 9 (5 + `53 / 2)` = 9 (`58 / 2)` = 9 (29) = 261
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