Number Series

Differences: 68-54=14, 85-68=17, 106-85=21, 134-106=28, 164-134=30. The pattern of differences should be 14, 17, 20, 23, 26. 134 is incorrect because 106+23=129 and 129+26=155.

The intended pattern is multiply by successive integers and add successive half-integers: x1 + 1.5, x2 + 2.5, x3 + 3.5, and so on. After 128, the next term should be 128 x 4 + 4.5 = 516.5, so 513.5 is the wrong term. The later terms match the corrected pattern.

The pattern of the series is that each term is obtained by multiplying the previous term by 2 and adding 1. Working backward from 9, we solve the equation 2x + 1 = 9, which gives x = 4.

Arithmetic progression: a = 7, d = -4. 19th term = a + (n-1)d = 7 + (18)(-4) = 7 - 72 = -65.

a12 = a + 11d, a5 = a + 4d. a12 = a5 + 14 implies 7d = 14, so d = 2. Sum: a12 + a5 = 36 implies (a + 11d) + (a + 4d) = 36, so 2a + 15d = 36. Substituting d=2: 2a + 30 = 36, 2a = 6, a = 3.

This is an arithmetic progression with first term a = 4 and common difference d = 5. The formula for the n-th term is a + (n-1)d = 104. So, 4 + (n-1)5 = 104, which simplifies to (n-1)5 = 100, n-1 = 20, n = 21.

The pattern is: 2*2+1=5, 5*2+2=12, 12*2+3=27, 27*2+4=58.

The differences are: 195-175=20, 175-150=25, 150-120=30. The next difference should be 35. 120-35 = 85.

The pattern follows alternating operations: multiply by 1.5, then divide by 3. Specifically: 36 * 1.5 = 54, 54 / 3 = 18, 18 * 1.5 = 27, 27 / 3 = 9, 9 * 1.5 = 13.5. The term 18.5 is incorrect because the sequence should continue with 13.5, then 13.5 / 3 = 4.5.

The series is 2^3, 3^3, 5^3, 7^3. These are cubes of consecutive prime numbers. The next prime is 11, and 11^3 = 1331.

Pattern: 7*2+6=20, 20*2+6=46, 46*2+6=98, 98*2+6=202, 202*2+6=410.