Summation notation ek compact aur smart tarika hai numbers ke sums ko express karne ka.
Ise mathematics ke bahut saare areas mein use kiya jaata hai, jaise calculus, statistics, aur probability.
Is article mein, hum summation notation aur uski properties introduce karenge.
Hum yeh bhi examples denge ki summation notation ka use mathematical problems solve karne ke liye kaise kiya jaa sakta hai.
What is Summation Notation?
Mathematics mein, ek well-defined tareeka hai jisse numbers ke sum ko compressed way mein likha jaa sakta hai.
Greek letter sigma (Σ) use kiya jaata hai summation represent karne ke liye.
Summation symbol ke baad teen parts aate hain: lower limit, upper limit, aur expression jo summed honi hai, jaise:
i=mΣn h(n)
- “i” = Lower limit woh pehla number hai jo sum mein shaamil hai.
- “n” = Upper limit woh aakhri number hai jo sum mein shaamil hai.
- “h(n)” = Expression jo summed honi hai woh same number hai jo repeatedly add kiya jaata hai.
Summation notation ka use kisi bhi size ke sums ko express karne ke liye kiya jaa sakta hai. Iske
alawa ise simple numbers se zyada complex expressions sum karne ke liye bhi use kiya jaa sakta hai. Example ke liye, yeh expression numbers 3 se 12 tak ke squares ko sum karta hai:
i=3Σ12 [i]2
Is expression ko padhna “i equals 3 se i equals 12 tak i squared ka sum” ke taur par padha jaa sakta hai. Dusre lafzon mein, yeh expression inn numbers ko sum karta hai:
i=3Σ12 [i]2 = 32 + 42 + 52 + 62 + 72 + 82 + 92 + 102 + 112 + 122
Summation Notation ki Properties
Summation notation ki bahut saari properties hain. Yahan se kuch sabse common properties hain:
Linearity Property
- For addition
Aap ek summation of sum of two sequences ko do alag summations ke sum mein split kar sakte ho:
i=mΣn [ai + bi] = i=mΣn [ai] + i=mΣn [bi]
- For subtraction
Aap ek summation of difference of two sequences ko do alag summations ke difference mein split kar sakte ho:
i=mΣn [ai – bi] = i=mΣn [ai] – i=mΣn [bi]
Scalar Multiplication Property
Ek constant ko ek summation se factor out kiya jaa sakta hai is property ka use karke, jaise:
i=mΣn [K ai] = K i=mΣn [ai]
Constant Property
i=mΣn [K] = K + K + K + K + … + K (n times)
Change of Index Property
Yeh property batati hai ki index variable ko change karne se sum ka value affect nahi hoga, jaise:
i=mΣn [ai] = j=mΣn [aj]
Shifting the Limits Property
Yeh property batati hai ki summation ke limits ko ek constant se shift kiya jaa sakta hai, jaise:
i=m+kΣn+k [ai-k]
Product Property
i=mΣn [ai x bi] = i=mΣn [ai] x i=mΣn [bi]
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Applications of Summation Notation
| Applications | Explanation |
| In Calculus | Summation notation is used to express sums of infinite series, which are used to represent functions and solve differential equations. |
| In Statistics | Summation notation is used to express sums of data, which are used to calculate statistical measures such as the mean and standard deviation. |
| In Linear Algebra | Summation notation is used to express sums of vectors and matrices, which are used to represent linear transformations. |
| In Probability | Summation notation is used to express sums of probabilities, which are used to calculate the probability of events. |
| In Combinatorics | Summation notation is used to express sums of combinations, which are used to count the number of possible arrangements of objects. |
| In Numerical Analysis | Numerical analysis: Summation notation is used to express sums of numerical approximations, which are used to approximate the values of functions. |
How to Solve the Problems?
- Problem ko samjho. Kaun si expression summed ho rahi hai? Sum ke lower aur upper limits kya hain?
- Expression jo summed ho rahi hai usko simplify karo. Iske liye summation notation ki properties use karni pad sakti hain.
- Sum ko evaluate karo. Iske liye kisi formula ya summation calculator ka use kar sakte ho.
Yahan kuch specific examples hain ki summation notation problems kaise solve kiye jaate hain:
Example 1: Rules ka use karke
Niche di gayi summation function ka sum calculate karo agar summation notation ka upper limit 21 hai aur lower limit 11 hai.
f(i) = 12i2 – 2i + 30
Solution
Step 1: Pehle, di gayi function ko likho aur usey summation notation ke general form ke hisaab se limits ke saath likho.
i=mΣnf(i) = i=11Σ21 [12i2 – 2i + 30]
Step 2: Ab upar di gayi expression ko addition aur subtraction ki linearity property ka use karke simplify karo.
i=11Σ21 [12i2 – 2i + 30] = i=11Σ21 [12i2] – i=11Σ21 [2i] + i=11Σ21 [30] i=11Σ21 [12i2 – 2i + 30] = 12i=11Σ21 [i2] – 2i=11Σ21 [i] + i=11Σ21 [30]
Step 3: Ab har term ke sum ko alag se evaluate karo.
For 12i=11Σ21 [i2]
12i=11Σ21 [i2] = 12[112 + 122 + 132 + 142 + 152 + 162 + 172 + 182 + 192 + 202 + 212] 12i=11Σ21 [i2] = 12[121 + 144 + 169 + 196 + 225 + 256 + 289 + 324 + 361 + 400 + 441] 12i=11Σ21 [i2] = 12[2926] 12i=11Σ21 [i2] = 35112
For 2i=11Σ21 [i]
2i=11Σ21 [i] = 2 [11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21] 2i=11Σ21 [i] = 2 [176] 2i=11Σ21 [i] = 352
For i=11Σ21 [30]
i=11Σ21 [30] = 30 + 30 + 30 + 30 + 30 + 30 + 30 + 30 + 30 + 30 (constant rule se) i=11Σ21 [30] = 30 * 10 i=11Σ21 [30] = 300
Step 4: Ab upar ke values ko equation mein daal kar di gayi expression ka sum calculate karo.
i=11Σ21 [12i2 – 2i + 30] = 12i=11Σ21 [i2] – 2i=11Σ21 [i] + i=11Σ21 [30] i=11Σ21 [12i2 – 2i + 30] = 35112 – 352 + 300
i=11Σ21 [12i2 – 2i + 30] = 34760 + 300 i=11Σ21 [12i2 – 2i + 30] = 35060
Example 2: Values place karke
Di gayi summation function ka sum calculate karo agar summation notation ka upper limit 9 hai aur lower limit 4 hai.
f(j) = 4j2 + 2j3 – 3j
Solution
Step 1: Pehle, di gayi function ko likho aur usey summation notation ke general form ke hisaab se limits ke saath likho.
j=mΣnf(j) = j=4Σ9 [4j2 + 2j3 – 3j]
Step 2: Ab upar di gayi expression mein 4 se 9 tak ke values daal kar simplify karo.
For j = 4 4j2 + 2j3 – 3j = 4(4)2 + 2(4)3 – 3(4) 4j2 + 2j3 – 3j = 4(16) + 2(64) – 3(4) 4j2 + 2j3 – 3j = 64 + 128 – 12 4j2 + 2j3 – 3j = 180
For j = 5 4j2 + 2j3 – 3j = 4(5)2 + 2(5)3 – 3(5) 4j2 + 2j3 – 3j = 4(25) + 2(125) – 3(5) 4j2 + 2j3 – 3j = 100 + 250 – 15 4j2 + 2j3 – 3j = 335
For j = 6 4j2 + 2j3 – 3j = 4(6)2 + 2(6)3 – 3(6) 4j2 + 2j3 – 3j = 4(36) + 2(216) – 3(6)
4j2 + 2j3 – 3j = 144 + 432 – 18 4j2 + 2j3 – 3j = 558
For j = 7 4j2 + 2j3 – 3j = 4(7)2 + 2(7)3 – 3(7) 4j2 + 2j3 – 3j = 4(49) + 2(343) – 3(7) 4j2 + 2j3 – 3j = 196 + 686 – 21 4j2 + 2j3 – 3j = 861
For j = 8 4j2 + 2j3 – 3j = 4(8)2 + 2(8)3 – 3(8) 4j2 + 2j3 – 3j = 4(64) + 2(512) – 3(8) 4j2 + 2j3 – 3j = 256 + 1024 – 24 4j2 + 2j3 – 3j = 1256
For j = 9 4j2 + 2j3 – 3j = 4(9)2 + 2(9)3 – 3(9) 4j2 + 2j3 – 3j = 4(81) + 2(729) – 3(9) 4j2 + 2j3 – 3j = 324 + 1458 – 27 4j2 + 2j3 – 3j = 1755
Step 4: Ab upar saare results ko add kar ke di gayi expression ka sum calculate karo.
j=4Σ9 [4j2 + 2j3 – 3j] = 180 + 335 + 558 + 861 + 1256 + 1755 j=4Σ9 [4j2 + 2j3 – 3j] = 4945
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Wrap Up
Summation notation ek powerful method hai jo mathematical expressions ko simplify aur clarify karne ke liye use kiya ja sakta hai.
Iska use bahut saare areas of mathematics mein kiya jaata hai, jaise calculus, statistics, probability, linear algebra, combinatorics, aur numerical analysis. Is article mein, humne summation notation aur uski properties introduce ki hain.
Humne yeh bhi examples diye hain ki summation notation ka use mathematical problems solve karne ke liye kaise kiya ja sakta hai.
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