Number Series

The series follows a pattern of concatenating consecutive odd numbers: 1-3, 3-5, 5-7, 7-9, 9-11. Following this logic, the next pair after 9-11 is 11-13, which results in 1113.

The pattern is: 2*2+1=5, 5*2+3=13, 13*2+5=31, 31*2+7=69. The next term is 69*2+9 = 138+9 = 147.

Differences: 34-12=22, 74-34=40, 138-74=64. Second differences: 40-22=18, 64-40=24. The second difference increases by 6. Next second difference is 30. Next first difference is 64+30=94. Next term is 138+94=232.

The series follows a pattern of subtracting 103 from the previous term. 721 - 103 = 618, 618 - 103 = 515, 515 - 103 = 412, 412 - 103 = 309, and 309 - 103 = 206.

The pattern is: 6*2+1=13, 13*2-1=25, 25*2+1=51, 51*2-1=101, 101*2+1=203.

This is an arithmetic progression with a = 1 and d = 6. The sum S_n = (n/2) * [2a + (n-1)d]. 154 = (n/2) * [2(1) + (n-1)6] = (n/2) * [2 + 6n - 6] = (n/2) * [6n - 4] = 3n^2 - 2n. 3n^2 - 2n - 154 = 0. Solving for n: (3n + 20)(n - 7) = 0. So n = 7.

The series is 1, 1, 2, 4, 3, 9, 4, ... This is an interleaved series. The odd positions are 1, 2, 3, 4. The even positions are 1^2, 2^2, 3^2, 4^2. The next term is 4^2 = 16.

The series 2, 3, 5, 7 is a sequence of consecutive prime numbers. The next prime number after 7 is 11.

Pattern: 4368 - 3257 = 1111. 5477 - 4368 = 1109. 6588 - 5477 = 1111. The pattern alternates +1111, +1109. Next is 6588 + 1109 = 7697.

The sequence is 9, 11, 13, 15, ... This is an arithmetic progression with first term a = 9 and common difference d = 2. The n-th term is a + (n-1)d = 9 + (n-1)2 = 9 + 2n - 2 = 2n + 7.

The given sequence is an arithmetic progression with a first term of 8 and a common difference of 3. Using the formula for the n-th term, a_n = a + (n - 1)d, the 27th term is calculated as 8 + (27 - 1) * 3 = 86.

The sequence is an arithmetic progression with a common difference of 4. The next term is 12 + 4 = 16.

This is an arithmetic progression with a=3 and d=4. The 15th term is a + (15-1)d = 3 + 14*4 = 3 + 56 = 59.

Arithmetic sequence: a = 10, d = 5. 20th term = a + 19d = 10 + 19*5 = 10 + 95 = 105.