Key Concepts and Formulas

• Method of Differences: For a sequence where the differences between consecutive terms form a simpler sequence, this method helps determine the general term. If the second differences are constant, the general term is a quadratic polynomial of the form $T_r = Ar^2 + Br + C$.
• Summation Formulas:
  • Sum of first $n$ natural numbers: $\sum_{r=1}^{n} r = \frac{n(n+1)}{2}$
  • Sum of squares of first $n$ natural numbers: $\sum_{r=1}^{n} r^2 = \frac{n(n+1)(2n+1)}{6}$
  • Sum of a constant: $\sum_{r=1}^{n} c = nc$
• Linearity of Summation: $\sum_{r=1}^{n} (a \cdot f(r) + b \cdot g(r)) = a \sum_{r=1}^{n} f(r) + b \sum_{r=1}^{n} g(r)$

Step-by-Step Solution

Step 1: Analyze the Sequence and Find the General Term ($T_r$) The given series is $S = 1+3+11+25+45+71+\ldots$. To find the sum, we first need to determine the general term $T_r$. We examine the differences between consecutive terms: Terms: $1, 3, 11, 25, 45, 71, \ldots$ First Differences: $3-1=2, 11-3=8, 25-11=14, 45-25=20, 71-45=26, \ldots$ Second Differences: $8-2=6, 14-8=6, 20-14=6, 26-20=6, \ldots$ Since the second differences are constant (equal to 6), the general term $T_r$ is a quadratic in $r$, i.e., $T_r = Ar^2 + Br + C$.

We can relate the coefficients $A, B, C$ to the terms and differences. For a quadratic sequence $T_r = Ar^2 + Br + C$: The first term ($r=1$) is $T_1 = A+B+C$. The first difference between $T_1$ and $T_2$ is $T_2-T_1 = (4A+2B+C) - (A+B+C) = 3A+B$. The second difference is $(T_3-T_2) - (T_2-T_1) = (5A+B) - (3A+B) = 2A$.

From our sequence: $T_1 = 1$ First difference for $r=1$ (i.e., $T_2-T_1$) is 2. Second difference is 6.

Equating these with the general formulas: $2A = 6 \implies A = 3$. $3A+B = 2 \implies 3(3)+B = 2 \implies 9+B = 2 \implies B = -7$. $A+B+C = 1 \implies 3+(-7)+C = 1 \implies -4+C = 1 \implies C = 5$.

Therefore, the general term is $T_r = 3r^2 - 7r + 5$.

Let's verify for $r=1, 2, 3$: $T_1 = 3(1)^2 - 7(1) + 5 = 3 - 7 + 5 = 1$ (Correct) $T_2 = 3(2)^2 - 7(2) + 5 = 12 - 14 + 5 = 3$ (Correct) $T_3 = 3(3)^2 - 7(3) + 5 = 27 - 21 + 5 = 11$ (Correct)

Step 2: Calculate the Sum of the Series up to 20 Terms We need to find $S_{20} = \sum_{r=1}^{20} T_r$. $S_{20} = \sum_{r=1}^{20} (3r^2 - 7r + 5)$ Using the linearity of summation: $S_{20} = 3 \sum_{r=1}^{20} r^2 - 7 \sum_{r=1}^{20} r + 5 \sum_{r=1}^{20} 1$

Now, we apply the standard summation formulas with $n=20$:

1. $\sum_{r=1}^{20} r^2 = \frac{20(20+1)(2 \cdot 20+1)}{6} = \frac{20 \cdot 21 \cdot 41}{6} = \frac{420 \cdot 41}{6} = 70 \cdot 41 = 2870$.
2. $\sum_{r=1}^{20} r = \frac{20(20+1)}{2} = \frac{20 \cdot 21}{2} = 10 \cdot 21 = 210$.
3. $\sum_{r=1}^{20} 1 = 20$.

Substitute these values back into the expression for $S_{20}$: $S_{20} = 3(2870) - 7(210) + 5(20)$ $S_{20} = 8610 - 1470 + 100$ $S_{20} = 7140 + 100$ $S_{20} = 7240$

Common Mistakes & Tips

• Incorrectly identifying the type of sequence: Always start by calculating differences to determine if the sequence is arithmetic, geometric, quadratic, etc.
• Errors in solving for coefficients: Carefully solve the system of linear equations for $A, B, C$. A quick check of the general term with the first few terms is highly recommended.
• Calculation errors in summation formulas: Ensure you correctly substitute $n$ into the summation formulas and perform the arithmetic accurately.

Summary The given series has a general term that is a quadratic in $r$, determined by analyzing the differences between consecutive terms. After finding the general term $T_r = 3r^2 - 7r + 5$, the sum of the series up to 20 terms was calculated by applying standard summation formulas for powers of $r$ and using the linearity property of summation. The computed sum is 7240.

The final answer is $\boxed{7240}$.