Sequences and Series
- The sum (5³ + 6³ + .... 10³) is equal to :
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According to question ,
Sum = 5³ + 6³ + .... 10³
Sum = 125 + 216 + 343 + 512 + 729 + 1000 = 2925
Second method to solve this question :
Sn = (5³ + 6³ + ............ 10³)
Sn = (1³ + 2³ + 3³ + 4³ + 5³ +... 10³) – (1³ + 2³ + 3³ + 4³)Sn = 
n(n + 1) 
² - (1 + 8 + 27 + 64) 2
Correct Option: D
According to question ,
Sum = 5³ + 6³ + .... 10³
Sum = 125 + 216 + 343 + 512 + 729 + 1000 = 2925
Second method to solve this question :
Sn = (5³ + 6³ + ............ 10³)
Sn = (1³ + 2³ + 3³ + 4³ + 5³ +... 10³) – (1³ + 2³ + 3³ + 4³)Sn = n(n + 1) ² - (1 + 8 + 27 + 64) 2 Sn = 10(10 + 1) ² - 100 2
Sn = (55)² – 100 = 3025 – 100 = 2925
- If 1³ + 2³ + 3³ + .... + 10³ = 3025, then find the value of 2³ + 4³ + 6³ + .... + 20³
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Given in question ,
∵ 1³ + 2³ + 3³ + .... + 10³ = 3025 .....( 1 )
∴ 2³ + 4³ + 6³ + .... + 20³ = (2 × 1)³ + (2 × 2)³ + (2 × 3)³ + ..... + (2 × 10)³
Required answer = 8 × 1³ + 8 × 2³ + 8 × 3³ .... + 8 × 10³Correct Option: D
Given in question ,
∵ 1³ + 2³ + 3³ + .... + 10³ = 3025 .....( 1 )
∴ 2³ + 4³ + 6³ + .... + 20³ = (2 × 1)³ + (2 × 2)³ + (2 × 3)³ + ..... + (2 × 10)³
Required answer = 8 × 1³ + 8 × 2³ + 8 × 3³ .... + 8 × 10³
Required answer = 8 × [1³ + 2³ + 3³ + 4³ + ... + 10³]
Required answer = 8 × 3025 = 24200 { ∴
From ( 1 ) }
- If 1³ +2³ +3³ +4³ +5³ + 6³ = 441 then find the value of 2³ + 4³ + 6³ + 8³ + 10³ + 12³
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Given Here , 1³ + 2³ + 3³ + 4³ + 5³ + 6³ = 441 .....( 1 )
Now , 2³ + 4³ + 6³ + 8³ + 10³ + 12³
Required answer = 8 (1³ + 2³ + 3³ + 4³ + 5³ + 6³)Correct Option: D
Given Here , 1³ + 2³ + 3³ + 4³ + 5³ + 6³ = 441 .....( 1 )
Now , 2³ + 4³ + 6³ + 8³ + 10³ + 12³
Required answer = 8 (1³ + 2³ + 3³ + 4³ + 5³ + 6³)
Required answer = 8 × 441 = 3528 { Using ( 1 ) }
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If 1² + 2² + 3² ...... + x² = = x(x + 1) (2x + 1) then 1² + 3² + 5² + ..... 19² is equal to 2
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From the question ,
1² + 2² + 3² + ..... + n² = n(n + 1)(2n + 1) 6
∴ 1² + 3² + 5² + ..... + 19² = (1² + 2² + 3² + ..... + 20²) – (2² + 4² + ..... + 20²)∴ 1² + 3² + 5² + ..... + 19² = 20(20 + 1)(40 + 1) – 2² (1² + 2² + ..... + 10²) 6
Correct Option: A
From the question ,
1² + 2² + 3² + ..... + n² = n(n + 1)(2n + 1) 6
∴ 1² + 3² + 5² + ..... + 19² = (1² + 2² + 3² + ..... + 20²) – (2² + 4² + ..... + 20²)∴ 1² + 3² + 5² + ..... + 19² = 20(20 + 1)(40 + 1) – 2² (1² + 2² + ..... + 10²) 6 Required answer = 20 × 21 × 41 - 4 × 10 (10 + 1)(20 + 1) 6 6
Required answer = 2870 – 1540 = 1330
- If 1³ + 2³ + .... + 9³ = 2025, then the value of (0.11)³ + (0.22)³ + .... + (0.99)³ is close to
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Given , 1³ + 2³ + ..... + 9³ = 2025 ....( 1 )
Now, (0.11)³ + (0.22)³ + .... + (0.99)³ = 
11 
³ + 22 ³ +........+ 99 ³ 100 100 100 Required answer = 11 ³ (1³ + 2³ +......+ 9³) 100
Correct Option: B
Given , 1³ + 2³ + ..... + 9³ = 2025 ....( 1 )
Now, (0.11)³ + (0.22)³ + .... + (0.99)³ = 11 ³ + 22 ³ +........+ 99 ³ 100 100 100 Required answer = 11 ³ (1³ + 2³ +......+ 9³) 100 Required answer = 1331 × 2025 { Using ( 1 ) } 1000000 Required answer = 2695275 = 2.695275 ≈ 2.695 1000000
