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![🏆పోటీ పరీక్షల స్పెషల్ - SUCCESS PATH SOLUTION: LIMIT EVALUATED Evaluate the following limit: 3 2 7+) n lim + २ + n१ + ३ + . + m2 +1 +00 n2 1.ANALYZE THE TERMS k Sp = ) Define the sum as Sn m२ + k k=1 2. ESTABLISH BOUNDS Lower Bound (Ln) Upper Bound (U) k k Un = _ Ln = n2 + 1] 72 + T k=1 =1 [n(7ಯ 1)] 7(7 + 1) [26+4]| 1 1 72 + 1 72 + Tu २n२ + २ 2 3. APPLY SQUEEZE THEOREM Since both bounds approach the same limit as " _, ೦೦: m2 +m 1 lim Un = lim lim In lim ా 2 2n2 + 2 2 2 n-00 1+೦೦ 00 00 4. FINAL RESULT 1 The Limit is: 2 DQv Your Path to Success SUCCESS PATH SOLUTION: LIMIT EVALUATED Evaluate the following limit: 3 2 7+) n lim + २ + n१ + ३ + . + m2 +1 +00 n2 1.ANALYZE THE TERMS k Sp = ) Define the sum as Sn m२ + k k=1 2. ESTABLISH BOUNDS Lower Bound (Ln) Upper Bound (U) k k Un = _ Ln = n2 + 1] 72 + T k=1 =1 [n(7ಯ 1)] 7(7 + 1) [26+4]| 1 1 72 + 1 72 + Tu २n२ + २ 2 3. APPLY SQUEEZE THEOREM Since both bounds approach the same limit as " _, ೦೦: m2 +m 1 lim Un = lim lim In lim ా 2 2n2 + 2 2 2 n-00 1+೦೦ 00 00 4. FINAL RESULT 1 The Limit is: 2 DQv Your Path to Success - ShareChat](https://cdn4.sharechat.com/bd5223f_s1w/compressed_gm_40_img_669510_3492162a_1782405667280_sc.jpg?tenant=sc&referrer=user-profile-service%2FrequestType50&f=280_sc.jpg "🏆పోటీ పరీక్షల స్పెషల్ - SUCCESS PATH SOLUTION: LIMIT EVALUATED Evaluate the following limit: 3 2 7+) n lim + २ + n१ + ३ + . + m2 +1 +00 n2 1.ANALYZE THE TERMS k Sp = ) Define the sum as Sn m२ + k k=1 2. ESTABLISH BOUNDS Lower Bound (Ln) Upper Bound (U) k k Un = _ Ln = n2 + 1] 72 + T k=1 =1 [n(7ಯ 1)] 7(7 + 1) [26+4]| 1 1 72 + 1 72 + Tu २n२ + २ 2 3. APPLY SQUEEZE THEOREM Since both bounds approach the same limit as \" _, ೦೦: m2 +m 1 lim Un = lim lim In lim ా 2 2n2 + 2 2 2 n-00 1+೦೦ 00 00 4. FINAL RESULT 1 The Limit is: 2 DQv Your Path to Success SUCCESS PATH SOLUTION: LIMIT EVALUATED Evaluate the following limit: 3 2 7+) n lim + २ + n१ + ३ + . + m2 +1 +00 n2 1.ANALYZE THE TERMS k Sp = ) Define the sum as Sn m२ + k k=1 2. ESTABLISH BOUNDS Lower Bound (Ln) Upper Bound (U) k k Un = _ Ln = n2 + 1] 72 + T k=1 =1 [n(7ಯ 1)] 7(7 + 1) [26+4]| 1 1 72 + 1 72 + Tu २n२ + २ 2 3. APPLY SQUEEZE THEOREM Since both bounds approach the same limit as \" _, ೦೦: m2 +m 1 lim Un = lim lim In lim ా 2 2n2 + 2 2 2 n-00 1+೦೦ 00 00 4. FINAL RESULT 1 The Limit is: 2 DQv Your Path to Success - ShareChat")
![➗10th గణితం - ALGEBRA FORMULA HANDBOOK PART1: 30 Essentiol Alebraic_Identities for Students 1. BASIC IDENTITIES 2. CUBIC IDENTITIES (4 + b)? = 42 + 24b + b2 (a+ 6)3 342b + 34b? + b3 a3 + (4 - b) 2 a2 _ Zab (a+6)3 b2 43 + b3 + 34b (4+ b) 42 + b2 = (4 + b) 2' (4 - b)3 43 _ 342b + 34b2 ' 03 Zab a2 _ 62 = (a+b) (a-b) (-b)3 43 _ b3 _ 3.b (4 - b) (4 - b) 2 (4 + b) 2 43 + b3 = (4+b) ( 42 Aab 0b + 02) ( - b)2 02 + 02 = + 2ab (a-b) (a? + ab + B2) - b3 0 2(02 + b2) . (a + b)२ + (a - b)2 43 + b3 = (4 + b)3 3ab (a+b) (a+6)2 (a-6)2 44b _ b3 (a-b)3 + 3ab (a-b) (4x೬)? - (42೬)2 ab 3. IDENTITIES INVOLVING THREE VARIABLES (a+ b+c)2 42 + b2 + c2 + 2 (4b + bc + c4 ) = a? + b2+c2 = (a+b+c)2 _ 2 (ab a bc + ca)| a (a+b+c)2 _ (a2+ 62+c2) 2(ab bc + ca) (a+b+c)3 43 + b3 + c3 + 3 (4 + b) (b + c) (c +4) 43 + b3 + c3 ' (a+b+c) (a?| ४२ + ८२ 3abc ab - bc ca) ३ (a + b + c) [ (a - b)२ ৮3 + c3 (b - c)२ (c-a)2] a 3 3abc 4. PRODUCT IDENTITIES 5. SPECIAL RESULTS (x+a)(r+b) = x2 (a+b)x + ab bc(b-c)+ ca(c-a)+ ab(a-b) (b - c) (c - &) (4 - b) (xta) (x-b) = x2 + (a-b)x-abl a२ (b - c) + b२ (c - a ) + c२ (a - b) (x-a) (x-b) = x2 (a+b)x+ab (b - c) (c - &) (4 - b) (X+p) (X+0) (X+r) c2) + b(c? _ 42) + c (4? _ b2) (P+9+r)x2 0(62 (Pq (b - c) (c - &) (4 - b) 4r + rP)x + Pqr Practice more Score higher Understand better ALGEBRA FORMULA HANDBOOK PART1: 30 Essentiol Alebraic_Identities for Students 1. BASIC IDENTITIES 2. CUBIC IDENTITIES (4 + b)? = 42 + 24b + b2 (a+ 6)3 342b + 34b? + b3 a3 + (4 - b) 2 a2 _ Zab (a+6)3 b2 43 + b3 + 34b (4+ b) 42 + b2 = (4 + b) 2' (4 - b)3 43 _ 342b + 34b2 ' 03 Zab a2 _ 62 = (a+b) (a-b) (-b)3 43 _ b3 _ 3.b (4 - b) (4 - b) 2 (4 + b) 2 43 + b3 = (4+b) ( 42 Aab 0b + 02) ( - b)2 02 + 02 = + 2ab (a-b) (a? + ab + B2) - b3 0 2(02 + b2) . (a + b)२ + (a - b)2 43 + b3 = (4 + b)3 3ab (a+b) (a+6)2 (a-6)2 44b _ b3 (a-b)3 + 3ab (a-b) (4x೬)? - (42೬)2 ab 3. IDENTITIES INVOLVING THREE VARIABLES (a+ b+c)2 42 + b2 + c2 + 2 (4b + bc + c4 ) = a? + b2+c2 = (a+b+c)2 _ 2 (ab a bc + ca)| a (a+b+c)2 _ (a2+ 62+c2) 2(ab bc + ca) (a+b+c)3 43 + b3 + c3 + 3 (4 + b) (b + c) (c +4) 43 + b3 + c3 ' (a+b+c) (a?| ४२ + ८२ 3abc ab - bc ca) ३ (a + b + c) [ (a - b)२ ৮3 + c3 (b - c)२ (c-a)2] a 3 3abc 4. PRODUCT IDENTITIES 5. SPECIAL RESULTS (x+a)(r+b) = x2 (a+b)x + ab bc(b-c)+ ca(c-a)+ ab(a-b) (b - c) (c - &) (4 - b) (xta) (x-b) = x2 + (a-b)x-abl a२ (b - c) + b२ (c - a ) + c२ (a - b) (x-a) (x-b) = x2 (a+b)x+ab (b - c) (c - &) (4 - b) (X+p) (X+0) (X+r) c2) + b(c? _ 42) + c (4? _ b2) (P+9+r)x2 0(62 (Pq (b - c) (c - &) (4 - b) 4r + rP)x + Pqr Practice more Score higher Understand better - ShareChat](https://cdn4.sharechat.com/bd5223f_s1w/compressed_gm_40_img_750173_1953fd9e_1781934716122_sc.jpg?tenant=sc&referrer=user-profile-service%2FrequestType50&f=122_sc.jpg "➗10th గణితం - ALGEBRA FORMULA HANDBOOK PART1: 30 Essentiol Alebraic_Identities for Students 1. BASIC IDENTITIES 2. CUBIC IDENTITIES (4 + b)? = 42 + 24b + b2 (a+ 6)3 342b + 34b? + b3 a3 + (4 - b) 2 a2 _ Zab (a+6)3 b2 43 + b3 + 34b (4+ b) 42 + b2 = (4 + b) 2' (4 - b)3 43 _ 342b + 34b2 ' 03 Zab a2 _ 62 = (a+b) (a-b) (-b)3 43 _ b3 _ 3.b (4 - b) (4 - b) 2 (4 + b) 2 43 + b3 = (4+b) ( 42 Aab 0b + 02) ( - b)2 02 + 02 = + 2ab (a-b) (a? + ab + B2) - b3 0 2(02 + b2) . (a + b)२ + (a - b)2 43 + b3 = (4 + b)3 3ab (a+b) (a+6)2 (a-6)2 44b _ b3 (a-b)3 + 3ab (a-b) (4x೬)? - (42೬)2 ab 3. IDENTITIES INVOLVING THREE VARIABLES (a+ b+c)2 42 + b2 + c2 + 2 (4b + bc + c4 ) = a? + b2+c2 = (a+b+c)2 _ 2 (ab a bc + ca)| a (a+b+c)2 _ (a2+ 62+c2) 2(ab bc + ca) (a+b+c)3 43 + b3 + c3 + 3 (4 + b) (b + c) (c +4) 43 + b3 + c3 ' (a+b+c) (a?| ४२ + ८२ 3abc ab - bc ca) ३ (a + b + c) [ (a - b)२ ৮3 + c3 (b - c)२ (c-a)2] a 3 3abc 4. PRODUCT IDENTITIES 5. SPECIAL RESULTS (x+a)(r+b) = x2 (a+b)x + ab bc(b-c)+ ca(c-a)+ ab(a-b) (b - c) (c - &) (4 - b) (xta) (x-b) = x2 + (a-b)x-abl a२ (b - c) + b२ (c - a ) + c२ (a - b) (x-a) (x-b) = x2 (a+b)x+ab (b - c) (c - &) (4 - b) (X+p) (X+0) (X+r) c2) + b(c? _ 42) + c (4? _ b2) (P+9+r)x2 0(62 (Pq (b - c) (c - &) (4 - b) 4r + rP)x + Pqr Practice more Score higher Understand better ALGEBRA FORMULA HANDBOOK PART1: 30 Essentiol Alebraic_Identities for Students 1. BASIC IDENTITIES 2. CUBIC IDENTITIES (4 + b)? = 42 + 24b + b2 (a+ 6)3 342b + 34b? + b3 a3 + (4 - b) 2 a2 _ Zab (a+6)3 b2 43 + b3 + 34b (4+ b) 42 + b2 = (4 + b) 2' (4 - b)3 43 _ 342b + 34b2 ' 03 Zab a2 _ 62 = (a+b) (a-b) (-b)3 43 _ b3 _ 3.b (4 - b) (4 - b) 2 (4 + b) 2 43 + b3 = (4+b) ( 42 Aab 0b + 02) ( - b)2 02 + 02 = + 2ab (a-b) (a? + ab + B2) - b3 0 2(02 + b2) . (a + b)२ + (a - b)2 43 + b3 = (4 + b)3 3ab (a+b) (a+6)2 (a-6)2 44b _ b3 (a-b)3 + 3ab (a-b) (4x೬)? - (42೬)2 ab 3. IDENTITIES INVOLVING THREE VARIABLES (a+ b+c)2 42 + b2 + c2 + 2 (4b + bc + c4 ) = a? + b2+c2 = (a+b+c)2 _ 2 (ab a bc + ca)| a (a+b+c)2 _ (a2+ 62+c2) 2(ab bc + ca) (a+b+c)3 43 + b3 + c3 + 3 (4 + b) (b + c) (c +4) 43 + b3 + c3 ' (a+b+c) (a?| ४२ + ८२ 3abc ab - bc ca) ३ (a + b + c) [ (a - b)२ ৮3 + c3 (b - c)२ (c-a)2] a 3 3abc 4. PRODUCT IDENTITIES 5. SPECIAL RESULTS (x+a)(r+b) = x2 (a+b)x + ab bc(b-c)+ ca(c-a)+ ab(a-b) (b - c) (c - &) (4 - b) (xta) (x-b) = x2 + (a-b)x-abl a२ (b - c) + b२ (c - a ) + c२ (a - b) (x-a) (x-b) = x2 (a+b)x+ab (b - c) (c - &) (4 - b) (X+p) (X+0) (X+r) c2) + b(c? _ 42) + c (4? _ b2) (P+9+r)x2 0(62 (Pq (b - c) (c - &) (4 - b) 4r + rP)x + Pqr Practice more Score higher Understand better - ShareChat")