Direction : Study the following information carefully and answer the given question. Observe the Series I carefully to identify the logic and obtain the value of 'P'. In both of the series, the same logic is applied. Series I: 60, 68, 14, (P), -294, 786, -1272 Series II: (P-12), Q, R, S, T, U, W If the average of the values of 'S' and 'U' will be equal to the first term of a newly formed third series, then which of the following statements is/are true? It is assumed that each of the series is following the same logic in this question. (i) The sum of the values of 'T' and 'W' is the multiple of 9. (ii) The fourth term of the third series is an odd number. (iii) The difference between the value of 'Q' and the second term of the third series is 426.

### Concept & Complex Number Series Extensions This question requires extending the logical pattern found in Series I (alternating addition and subtraction of $(n-1) \times n^3$) to fully flesh out Series II and then generate a completely new Series III to evaluate specific logical statements. $$ \text{Logic Pattern} = + (1 \times 2^3), - (2 \times 3^3), + (3 \times 4^3), - (4 \times 5^3), \dots $$ ### Step-by-Step Solution 1. **Complete Series II:** From the previous question, we know $P = 206$. * $T_1 = P - 12 = 194$ * $Q = 194 + 8 = 202$ * $R = 202 - 54 = 148$ * $S = 148 + (3 \times 4^3) = 148 + 192 = 340$ * $T = 340 - (4 \times 5^3) = 340 - 500 = -160$ * $U = -160 + (5 \times 6^3) = -160 + 1080 = 920$ * $W = 920 - (6 \times 7^3) = 920 - 2058 = -1138$ 2. **Form Series III:** * First Term = Average of S and U = $\frac{340 + 920}{2} = \frac{1260}{2} = 630$. * $T_1 = 630$ * $T_2 = 630 + (1 \times 2^3) = 630 + 8 = 638$ * $T_3 = 638 - (2 \times 3^3) = 638 - 54 = 584$ * $T_4 = 584 + (3 \times 4^3) = 584 + 192 = 776$ 3. **Evaluate the Statements:** * *(i) The sum of the values of 'T' and 'W' is the multiple of 9:* Sum = $-160 + (-1138) = -1298$. The sum of digits of 1298 is $1+2+9+8 = 20$, which is not divisible by 9. (False) * *(ii) The fourth term of the third series is an odd number:* The 4th term is 776, which is an even number. (False) * *(iii) The difference between the value of 'Q' and the second term of the third series is 426:* $Q = 202$, 2nd term of Series III = 638. Difference = $|202 - 638| = 436$. The statement says 426. (False) ### Exam Strategy & Shortcut For multi-layered series questions, carefully list out all calculated variables ($Q, R, S, T, U, W$) in a clean column. Because errors in early terms cascade and invalidate the entire question, double-check your initial arithmetic for $P$, $S$, and $U$ before testing the multiple-choice statements. ### Common Pitfall A frequent error is miscalculating the alternating signs for higher indices. For example, incorrectly adding the 6th index term instead of subtracting it will completely ruin the value of $W$ and any statement dependent on it. ### Final Answer Therefore, the correct answer is **None is true**.