Sequences and Series
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The sum of n terms the series 1 + 1 + 1 + 1 + ..... is 2 2² 2³
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As per the given series ,
1 + 1 + 1 + 1 + ....... n to n terms is a Geometric series 2 2² 2³
whose first term (a) is 1 and the commonratio (r) is
1 / 2 .
Using the given formula ,Sn = a(1 - rn) 1 - r Sn = 
1 - 1 
2n 1 - 1 2 Sn = 2n - 1 2n 1 2
Correct Option: A
As per the given series ,
1 + 1 + 1 + 1 + ....... n to n terms is a Geometric series 2 2² 2³
whose first term (a) is 1 and the commonratio (r) is
1 / 2 .
Using the given formula ,Sn = a(1 - rn) 1 - r Sn = 1 - 1 2n 1 - 1 2 Sn = 2n - 1 2n 1 2 Sn = 2. 2n - 1 = 2n - 1 2n 2n - 1
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1 - 1 1 - 1 1 - 1 ........... 1 - 1 is equal to 3 4 5 25
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From the question ,
Expression = 1 - 1 1 - 1 1 - 1 ..... 1 - 1 1 - 1 3 4 5 24 25 Expression = 2 × 3 × 4 ........× 23 × 24 3 4 5 24 25 Correct Option: A
From the question ,
Expression = 1 - 1 1 - 1 1 - 1 ..... 1 - 1 1 - 1 3 4 5 24 25 Expression = 2 × 3 × 4 ........× 23 × 24 = 2 3 4 5 24 25 25
- The sixth term of the sequence 2, 6, 11, 17, ..... is
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The pattern of following series is -
2 + 4 = 6
6 + 5 = 11
11 + 6 = 17
........... and so onCorrect Option: C
The pattern of following series is -
2 + 4 = 6
6 + 5 = 11
11 + 6 = 17
17 + 7 = 24
24 + 8 = 32
Therefore , The sixth term of the sequence is 32 .
- The ratio of the fifth and sixth terms of the sequence 1, 3, 6, 10, ...... is
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The pattern of the sequence is :
First term = 1
Second term = 1 + 2 = 3
Third term = 3 + 3 = 6
Fourth term = 6 + 4 = 10
Fifth term = 10 + 5 = 15
Sixth term = 15 + 6 = 21
∴ Required ratio = fifth term : sixth termCorrect Option: B
The pattern of the sequence is :
First term = 1
Second term = 1 + 2 = 3
Third term = 3 + 3 = 6
Fourth term = 6 + 4 = 10
Fifth term = 10 + 5 = 15
Sixth term = 15 + 6 = 21
∴ Required ratio = fifth term : sixth term
∴ Required ratio = 15 : 21 = 5 : 7
- The middle term(s) of the following series 2 + 4 + 6 + ... + 198 is
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Given , 2 + 4 + 6 + 8 + ...... + 198 = 2 (1 + 2 + 3 + ....... + 99 )
∴ Number of terms = 99Middle term = 99 + 1 = 50th term 2
Middle term = 100
Second method to find the required answer with the help of formula :
It is an arithmetic series.
a = 2, an = 198, d = common difference = 2
Number of terms = n
∴ an = a + (n – 1)d
⇒ 198 = 2 + (n – 1)2
⇒ (n – 1)2 = 198 – 2 = 196⇒ n – 1 = 196 = 98 2
Correct Option: D
Given , 2 + 4 + 6 + 8 + ...... + 198 = 2 (1 + 2 + 3 + ....... + 99 )
∴ Number of terms = 99Middle term = 99 + 1 = 50th term 2
Middle term = 100
Second method to find the required answer with the help of formula :
It is an arithmetic series.
a = 2, an = 198, d = common difference = 2
Number of terms = n
∴ an = a + (n – 1)d
⇒ 198 = 2 + (n – 1)2
⇒ (n – 1)2 = 198 – 2 = 196⇒ n – 1 = 196 = 98 2
⇒ n = 99Middle term = 99 + 1 = 50th term 2
∴ a50 = 2 + (50 – 1)2
a50 = 2 + 98 = 100
