Sequences and Series
- Find the sum of the first five terms of the following series.
1 + 1 + 1 + .......... + ......... 1 × 4 4 × 7 7 × 10
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Here , Expression = 1 + 1 + 1 + 1 + 1 1 × 4 4 × 7 7 × 10 10 × 13 13 × 16 Expression = 
1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 
× 1 4 4 7 7 10 10 13 13 16 3
Correct Option: C
Here , Expression = 1 + 1 + 1 + 1 + 1 1 × 4 4 × 7 7 × 10 10 × 13 13 × 16 Expression = 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 × 1 4 4 7 7 10 10 13 13 16 3 Expression = 15 × 1 = 5 16 3 16
- If (1012 + 25)² – (1012 – 25)² = 10n, then the value of n is
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With the help of given below formula ,
(a + b)² - (a - b)² = 4ab
∴ (1012 + 25)² - (1012 - 25)² = 10n
Here , a = 1012 , b = 25
⇒ 4 × 1012 × 25 = 10n
⇒ 100 × 1012 = 10n
⇒ 102 × 1012 = 10nCorrect Option: B
With the help of given below formula ,
(a + b)² - (a - b)² = 4ab
∴ (1012 + 25)² - (1012 - 25)² = 10n
Here , a = 1012 , b = 25
⇒ 4 × 1012 × 25 = 10n
⇒ 100 × 1012 = 10n
⇒ 102 × 1012 = 10n
⇒ 1014 = 10n
Equating powers on both sides ,
⇒ n = 14
- Given 1 + 2 + 3 + 4 + .... + 10 = 55, then the sum 6 + 12 + 18 + 24 + .... + 60 is equal to :
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Given in question , 1 + 2 + 3 + 4 + .... + 10 = 55.
Then, 6 + 12 + 18 + 24 + .... + 60 = 6 (1 + 2 + 3 + 4 + .... + 10)Correct Option: C
Given in question , 1 + 2 + 3 + 4 + .... + 10 = 55.
Then, 6 + 12 + 18 + 24 + .... + 60 = 6 (1 + 2 + 3 + 4 + .... + 10)
∴ Required answer = 6 × 55 = 330
- When simplified the product
1 - 1 1 - 1 1 - 1 ........... 1 - 1 2 3 4 n
gives :
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Here , Expression = 1 - 1 1 - 1 1 - 1 ...........
1 - 1 2 3 4 n Expression = 1 × 2 × 3 × ..... × n - 1 2 3 4 n Correct Option: A
Here , Expression = 1 - 1 1 - 1 1 - 1 ...........
1 - 1 2 3 4 n Expression = 1 × 2 × 3 × ..... × n - 1 = 1 2 3 4 n n
- The value of
3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 is 1².2² 2².3² 3².4² 4².5² 5².6² 6².7² 7².8² 8².9² 9².10²
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As per the given question ,
Expression = 3 + 5 + 7 + .....+ 17 + 19 1².2² 2².3² 3².4² 8².9² 9².10² Expression = 2² - 1² + 3² - 2² + 4² - 3² + .....+ 9² - 8² + 10² - 9² 1².2² 2².3² 3².4² 8².9² 9².10² Expression = 1 - 1 + 1 - 1 + 1 - 1 +..... 1 - 1 + 1 - 1 1² 2² 2² 3² 3² 4² 8² 9² 9² 10² Required answer = 1 - 1 + 1 - 1 + 1 - 1 + ..... + 1 - 1 + 1 - 1 1² 2² 2² 3² 3² 4² 8² 9² 9² 10² Required answer = 1 - 1 10²
Correct Option: B
As per the given question ,
Expression = 3 + 5 + 7 + .....+ 17 + 19 1².2² 2².3² 3².4² 8².9² 9².10² Expression = 2² - 1² + 3² - 2² + 4² - 3² + .....+ 9² - 8² + 10² - 9² 1².2² 2².3² 3².4² 8².9² 9².10² Expression = 1 - 1 + 1 - 1 + 1 - 1 +..... 1 - 1 + 1 - 1 1² 2² 2² 3² 3² 4² 8² 9² 9² 10² Required answer = 1 - 1 + 1 - 1 + 1 - 1 + ..... + 1 - 1 + 1 - 1 1² 2² 2² 3² 3² 4² 8² 9² 9² 10² Required answer = 1 - 1 10² Required answer = 1 - 1 = 100 - 1 100 100 Required answer = 99 100
