Before going to learn summation formulas, first, we will recall the meaning of summation. Summation (or) amount is the sum of consecutive terms of a sequence. To compose the amount of more terms, say n terms, the a sequence \(\a_n\\), we use the summation notation rather of writing the entirety sum manually. I.e., \(a_1+a_2+...+a_n= \sum_i=1^n a_i\). Allow us find out the summation formulas and their applications making use of a couple of solved examples.
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What AreSummation Formulas?
The summation recipe are supplied to calculate the sum of thesequence. There room various species of order such together arithmetic sequence, geometric sequence, etc and also hence there room various species of summation recipe of different sequences. Also, there are summation formulas to discover the sum of the herbal numbers, the amount of squares of herbal numbers, the amount of cubes of organic numbers, the sum of also numbers, the sum of strange numbers, etc.Here isthe perform ofsummation formulas. We will find out each of these formulas in information in the upcoming section.
List ofSummation Formulas
We know that the amount of two numbers is the an outcome obtained by including two numbers. Thus, if \(\x_1, x_2,…,x_n\\) is a sequence, then the sum of its state is denoted using the symbolΣ (sigma). I.e., the amount of the above sequence =\(\sum_i=1^nx_i=x_1+x_2+….x_n\). Here, \(\sum_i=1^n\) represents the amount of the terms of the sequence from the 1stterm come the nthterm and also it is check out as "sigma i is same to 1 to n". But we actually perform not need to include the amount of the order manually all the time to uncover the sum. Instead, we usage the adhering to summation formulas. Below are some popular summation formulas.
The sum of the 4th powers of the an initial n organic numbers is calculated using the formula:\(\sum_i=1^n i^4\) = 14+ 24+ 34+ ... + n4= \(\dfrac130 n(n+1)(2 n+1)\left(3 n^2+3 n-1\right)\)
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We will see the applications of the summation formulas in the upcoming section.
Examples UsingSummation Formulas
Example 1:Find the amount of all also numbers indigenous 1 to 100.
We recognize that the variety of even numbers from 1 to 100 is n = 50.
Using the summation formulas, the amount of the very first n even numbers is
n (n + 1) = 50 (50 + 1) = 50 (51) =2550
Answer:The required sum =2,550.
Example 2:Find the value of\(\sum_i=1^n (3-2i)\) making use of the summation formulas.
To find: The offered sum making use of the summation formulas.
\(\beginalign &\sum_i=1^n (3 -2i)\\<0.2cm>&= 3 \sum_i=1^n 1 - 2 \sum_i=1^n i\\<0.2cm> &= 3 n - 2 \left( \dfracn(n+1)2 \right)\\<0.2cm> &= \dfrac6n -2n^2-2n2\\<0.2cm> &= \dfrac4n-2n^22\\<0.2cm> &= 2n-n^2 \endalign\)
Answer:\(\sum_i=1^n (3 -2i)=2n-n^2\).
Example 3:Find the worth of the summation \(\sum_k=1^150(k-3)^2\)using the summation formulas.
To find:The provided sum utilizing the summation formulas.
\(\beginalign &\sum_k=1^150(k-3)^2 \\<0.2cm>&= \sum_k=1^150 (k^2 -6k+9)\\<0.2cm> &= \sum_k=1^150 k^2 - 6 \sum_k=1^150 k + 9 \sum_k=1^150 1 \\<0.2cm> &= \dfrac150(150+1)(2(150)+1)6- 6 \cdot \dfrac150(150+1)2 + 9 (150)\\<0.2cm> &= 1136275 -67950 + 1350\\<0.2cm> &=1069675 \endalign\)
Answer: \( \sum_k=1^150(k-3)^2\) =1,069,675.
FAQs onSummation Formulas
What Is the Summation Formula of organic Numbers?
To discover the amount of the natural numbers from 1 to n, we usage the formula n (n + 1) / 2. For example, the amount of the an initial 50 organic numbers is, 50 (50 + 1) / 2 =1275.
What space the Applications the the Summation Formulas?
The summation recipe are used to uncover the sum of any particular sequence without in reality finding the amount manually. Because that example, the summation formula of finding the amount of the first n weird number is n2. Using this, we have the right to say the the sum of the first 30 odd numbers is 12+ 32+ ... (30 numbers)= 302= 900.
What Is the general Summation Formula?
The basic summation formula states that the sum of a sequence\(\x_1, x_2,…,x_n\\) is denoted utilizing the symbolΣ. I.e., the amount of the above sequence =\(\sum_i=1^nx_i=x_1+x_2+….x_n\).
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How To derive the Summation Formula of strange Numbers?
Let us consider n odd numbers 1, 3, 5, ..., (2n + 1). Because the difference in between every 2 odd numbers is 2, this succession is arithmetic. Making use of the summation formula the arithmetic sequence, the amount of n odd numbers is n / 2 < 2+ (n - 1) 2> = n/2 < 2 + 2n - 2> = n/2 (2n) = n2. Thus, the sum of the very first n odd organic numbers is n2.