Problem 1 :
Find the value of
12 + 22 + 32 + ........ + 502
Problem 2 :
Find the value of
152 + 162 + 172 + ........ + 282
Problem 3 :
Find the value of
52 + 102 + 152 + ........ + 1052
Problem 4 :
Find the sum of
2 + 8 + 18 + ........ + to 30 terms
Problem 5 :
Find the sum of
3 + 12 + 27 + ........ + 1875
Problem 6 :
Find the average of squares first 25 natural numbers.
Problem 7 :
Rekha has 15 square color papers of sizes 10 cm, 11 cm, 12 cm,…, 24 cm. How much area can be decorated with these color papers?
Problem 8 :
The sum of first n natural numbers is 120, while the sum of their squares is 1240. Find the value of n.
1. Answer :
= 12 + 22 + 32 + ........ + 502
Using 12 + 22 + 32 + ........ + n2 = [n(n + 1)(2n + 1)]/6,
= [50(50 + 1)(2x50 + 1)]/6
= [50(50 + 1)(100 + 1)]/6
= [50(51)(101)]/6
= [50(51)(101)]/6
= 42925
2. Answer :
152 + 162 + 172 + ........ + 282 :
= (12 + 22 + 32 + ........ + 282) - (12 + 22 + 32 + ........ + 142)
Using 12 + 22 + 32 + ........ + n2 = [n(n + 1)(2n + 1)]/6,
= [28(28 + 1)(2x28 + 1)]/6 - [14(14 + 1)(2x14 + 1)]/6
= [28(29)(56 + 1)]/6 - [14(15)(28 + 1)]/6
= [28(29)(57)]/6 - [14(15)(29)]/6
= 7714 - 1015
= 6699
3. Answer :
= 52 + 102 + 152 + ........ + 1052
= (5x1)2 + (5x2)2 + (5x3)2 + ........ + (5x21)2
= 5212 + 5222 + 5232 + ........ + 52212
= 52(12 + 22 + 32 + ........ + 212)
= 25(12 + 22 + 32 + ........ + 212)
Using 12 + 22 + 32 + ........ + n2 = [n(n + 1)(2n + 1)]/6,
= 25[21(21 + 1)(2x21 + 1)]/6
= 25[21(22)(42 + 1)]/6
= 25[21(22)(43)]/6
= 82775
4. Answer :
2 + 8 + 18 + ........ + to 30 terms :
= 2(1 + 4 + 9 + ........ to 30 terms)
= 2(12 + 22 + 32 + ........ to 30 terms)
= 2(12 + 22 + 32 + ........ + 302)
Using 12 + 22 + 32 + ........ + n2 = [n(n + 1)(2n + 1)]/6,
= 2[30(30 + 1)(2x30 + 1)]/6
= 2[30(31)(60 + 1)]/6
= 2[30(31)(61)]/6
= 18910
5. Answer :
3 + 12 + 27 + ........ + 1875 :
= 3(1 + 4 + 9 + ........ + 625)
= 3(12 + 22 + 32 + ........ + 252)
Using 12 + 22 + 32 + ........ + n2 = [n(n + 1)(2n + 1)]/6,
= 3[25(25 + 1)(2x25 + 1)]/6
= 3[25(26)(50 + 1)]/6
= 3[25(26)(51)]/6
= 16575
6. Answer :
Using 12 + 22 + 32 + ........ + n2 = [n(n + 1)(2n + 1)]/6, to find the sum of squares first 25 natural numbers.
12 + 22 + 32 + ........ + 252 :
= [25(25 + 1)(2x25 + 1)]/6
= [25(26)(50 + 1)]/6
= [25(26)(51)]/6
= 5525
Average of squares first 25 natural numbers :
= (Sum of squares first 25 natural numbers)/25
= 5525/25
= 221
7. Answer :
Total area covered by the given 15 square color papers :
= 102 + 112 + 122 + ........ + 242
= (12 + 22 + 32 + ........ + 242) - (12 + 22 + 32 + ........ + 92)
Using 12 + 22 + 32 + ........ + n2 = [n(n + 1)(2n + 1)]/6,
= [24(24 + 1)(2x24 + 1)]/6 - [9(9 + 1)(2x9 + 1)]/6
= [24(25)(48 + 1)]/6 - [9(10)(18 + 1)]/6
= [24(25)(49)]/6 - [9(10)(19)]/6
= 4900 - 285
= 4615
The area of 4615 cm2 can be decorated with the 15 square color papers that Rekha has.
8. Answer :
Sum of first n natural numbers is 120 :
1 + 2 + 3 +, ...... + n = 120
[n(n + 1)]/2 = 120
Multiply each side by 2.
n(n + 1) = 240
Sum of their squares is 1240 :
12 + 22 + 32 + ........ + n2 = 1240
[n(n + 1)(2n + 1)]/6 = 1240
Substitute n(n + 1) = 240.
[240(2n + 1)]/6 = 1240
40(2n + 1) = 1240
Divide each side by 40.
2n + 1 = 31
Subtract 1 from each side.
2n = 30
Divide each side by 2.
n = 15
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