Directions (42-43):Given below three series I, II & III and each series has a wrong number . The number that should come in place of wrong number in series I, II & III is a, b & c respectively .

I. 5, 6, 8, 14, 38, 168, 878, 5918

II. 32, 544, 593, 809, 832, 898, 907

III. 18, 38, 124, 500, 2504, 15028, 105200

If $m^2=\sqrt{a+98}$, then which of the following statement/s is or are true.

(A) $(\frac{b}{m}+0.5)$ = resultant is an integer.

(B) $10m=c$

(C) $m+\frac{b-42}{18}\ge c$

To find the correct statements, we first need to identify the wrong number in each series and determine the correct values for a, b, and c.

Step 1: Analyze Series I
The given series is: 5, 6, 8, 14, 38, 168, 878, 5918
Let's find the difference between consecutive terms:
$6-5=1=1!$
$8-6=2=2!$
$14-8=6=3!$
$38-14=24=4!$
Following this factorial difference pattern, the next difference should be $5!=120$.
So, the term after 38 should be:
$38+120=158$
Let's check the subsequent terms using this corrected value:
$158+6!=158+720=878$
$878+7!=878+5040=5918$
Since these match the rest of the series, the wrong number is 168, and the correct number is a = 158.

Step 2: Analyze Series II
The given series is: 32, 544, 593, 809, 832, 898, 907
Let's examine the differences:
$544-32=512=8^3$
$593-544=49=7^2$
$809-593=216=6^3$
The pattern of differences is alternating cubes and squares of decreasing numbers: $8^3,7^2,6^3,5^2,4^3,3^2$.
So, the next difference should be $5^2=25$.
The term after 809 should be:
$809+25=834$
Let's verify with the remaining terms:
$834+4^3=834+64=898$
$898+3^2=898+9=907$
Since these match, the wrong number is 832, and the correct number is b = 834.

Step 3: Analyze Series III
The given series is: 18, 38, 124, 500, 2504, 15028, 105200
Let's analyze the relationship between consecutive terms starting from the later terms:
$15028\times 7+4=105196+4=105200$
$2504\times 6+4=15024+4=15028$
$500\times 5+4=2500+4=2504$
$124\times 4+4=496+4=500$
The pattern is $T_n=T_{n-1}\times n+4$.
Working backwards to the start of the series:
The second term should be:
$18\times 2+4=40$
Checking if 40 gives the next term:
$40\times 3+4=124$
This is correct. Thus, the wrong number is 38, and the correct number is c = 40.

Step 4: Find the value of m
We are given:
$m^2=\sqrt{a+98}$
Substituting a = 158:
$m^2=\sqrt{158+98}=\sqrt{256}=16$
Since $m^2=16$, m can be 4 or -4.

Step 5: Test the statements for both possible values of m

Case 1: If m = 4
• Statement (A):
$\frac{b}{m}+0.5=\frac{834}{4}+0.5=208.5+0.5=209$ (which is an integer) - True
• Statement (B):
$10m=10\times 4=40=c$ - True
• Statement (C):
$m+\frac{b-42}{18}=4+\frac{834-42}{18}=4+\frac{792}{18}=4+44=48$
Since $48\ge 40$ (which is c) - True

Case 2: If m = -4
• Statement (A):
$\frac{b}{m}+0.5=\frac{834}{-4}+0.5=-208.5+0.5=-208$ (which is an integer) - True
• Statement (B):
$10m=10\times (-4)=-40\ne 40$ - False
• Statement (C):
$m+\frac{b-42}{18}=-4+44=40$
Since $40\ge 40$ (which is c) - True

Because Statement (B) is not true when $m=-4$, only Statements (A) and (C) are true under all mathematically valid conditions for m.