Number Series

This is an arithmetic progression with first term a=8 and common difference d=4. The sum formula is S = n/2 * (2a + (n-1)d), so 476 = n/2 * (16 + (n-1)4). Solving 476 = 2n^2 + 6n leads to n^2 + 3n - 238 = 0, which factors to (n+17)(n-14) = 0, giving n=14.

This is the Fibonacci sequence where each term is the sum of the two preceding terms: 1+1=2, 1+2=3, 2+3=5, 3+5=8.

Series: 1, 2, 3, 5, 15, 30, 105. Pattern: 1*1=1, 1*2=2, 2*1.5=3, 3*2=6 (not 5), 6*2.5=15, 15*2=30, 30*3.5=105. The pattern is multiplying by 1, 1.5, 2, 2.5, 3, 3.5. The term 5 is wrong; it should be 6.

The differences are: 90-80=10, 80-73=7, 73-66=7, 66-60=6, 60-55=5, 55-51=4. The pattern of differences should be consistent. If 80 were 83, the differences would be 7, 7, 7, 6, 5, 4, which is more logical.

The pattern is 2^n - 1. 2^1-1=1, 2^2-1=3, 2^3-1=7, 2^4-1=15, 2^5-1=31, 2^6-1=63, 2^7-1=127. The number 33 is incorrect; it should be 31.

The pattern is multiplication by 4: 3*4=12, 12*4=48, 48*4=192, 192*4=768.

The series follows a pattern of multiplying by 1.5: 12 * 1.5 = 18; 18 * 1.5 = 27; 27 * 1.5 = 40.5; 40.5 * 1.5 = 60.75. The value 26.25 is incorrect; it should be 27.

The pattern is: 89 - 18 = 71; 71 - 16 = 55; 55 - 14 = 41; 41 - 12 = 29; 29 - 10 = 19. The difference decreases by 2 each time.

The series is an arithmetic progression with a = 90 and d = 3. The n-th term is 90 + (n-1)3 > 500. 3(n-1) > 410. n-1 > 136.66, so n > 137.66. The 138th term is 90 + 137*3 = 90 + 411 = 501. The next term is 504.

40n + 20 = 420. 40n = 400. n = 10.

The series is 1^2, 3^2, 6^2, 10^2, 15^2. The bases are 1, 3, 6, 10, 15. The differences between bases are 2, 3, 4, 5. The next base is 15 + 6 = 21. 21^2 = 441.

The series is an arithmetic progression with common difference d=2. Statement (1) says the middle term is 10. If there are n terms, the middle term is the average. This is insufficient without knowing the number of terms. Statement (2) says the first term is 2. With a=2 and d=2, the 2nd term is a+d = 2+2=4. Statement (2) is sufficient.