Let Tᵣ$T_r$ be the rth$r^{\text{th}}$ term of an A.P. If for some m, Tm = (1)/(25), T₂₅ = (1)/(20)$T_m = \frac{1}{25}, T_{25} = \frac{1}{20}$ and 20Σr = 1²⁵ Tᵣ = 13$20\sum_{r = 1}^{25} T_r = 13$ then 5mΣr = m2m Tᵣ$5m\sum_{r = m}^{2m} T_r$ is equal to:

Related Formula

Characteristic equation method for standard second-order linear homogeneous recurrence updates:

Core Logic

Solving the characteristic equation gives roots x = 1$x = 1$ and x = (3)/(2)$x = \frac{3}{2}$. The general solution takes the form:

Step 1: Evaluating Sequence Parameters

Using boundary conditions: For n = 0 A + B = 0$n = 0 \implies A + B = 0$ For n = 1 A + (3)/(2)B = (1)/(2)$n = 1 \implies A + \frac{3}{2}B = \frac{1}{2}$

Solving this simple linear system gives B = 1$B = 1$ and A = -1$A = -1$. Thus, the explicit sequence formula is:

Step 2: Summing the Target Range

Pattern Recognition

Characteristic roots directly decouple second-order linear loop progressions into basic combinations of clean geometric progressions.

Chapter Mix

Class 11 Maths: Sequences and Series

Related Formula

For a series whose consecutive differences form an Arithmetic Progression (A.P.), the general term is given by a quadratic expression:

Core Logic

Analyze the successive first-order differences of the terms:

Since the consecutive differences have a constant difference of 6$6$, they form an A.P. Set up the linear system for the first three terms:

• T₁ = a + b + c = 1$T_1 = a + b + c = 1$
• T₂ = 4a + 2b + c = 3$T_2 = 4a + 2b + c = 3$
• T₃ = 9a + 3b + c = 11$T_3 = 9a + 3b + c = 11$

Solving these equations simultaneously yields:

Step 1: Summing the Series

The general term is:

Evaluate the summation for n = 20$n = 20$ terms:

Substitute standard power sum formulas:

Pattern Recognition

Shortcut: When the first-order differences form an arithmetic sequence, the n$n$-th term is quadratic (an² + bn + c$an^2 + bn + c$) where the second difference is 2a$2a$ (2a = 6 a = 3$2a = 6 \implies a = 3$). Determine b$b$ and c$c$ using small values of n$n$, then apply standard Σ n²$\sum n^2$ and Σ n$\sum n$ summation formulas.

Evaluation Rubric / Model Answer

7240

Chapter Mix

Class 11 Mathematics: Sequences and Series

Related Formula

For a geometric sequence configuration with first term a$a$ and common ratio r$r$:

Core Logic

Convert information markers using parameter notations [cite: 1372, 1373, 1376]: a₃ a₅ = (ar²)(ar⁴) = a² r⁶ = 729 ar³ = 27$$a_3 a_5 = (ar^2)(ar^4) = a^2 r^6 = 729 \implies ar^3 = 27$$ [cite: 1373, 1374]

From second expression block data [cite: 1376]: a₂ + a₄ = ar + ar³ = (111)/(4)$$a_2 + a_4 = ar + ar^3 = \frac{111}{4}$$ [cite: 1376]

Substitute ar³ = 27$ar^3 = 27$ directly into the linear equation block [cite: 1376]: ar + 27 = (111)/(4) ar = (111)/(4) - 27 = (3)/(4)$$ar + 27 = \frac{111}{4} \implies ar = \frac{111}{4} - 27 = \frac{3}{4}$$ [cite: 1376]

Step 1: Finding parameters a and r

Divide the calculated components to evaluate the ratio [cite: 1387]: (ar³)/(ar) = (27)/(3/4) r² = 36 r = 6$$\frac{ar^3}{ar} = \frac{27}{3/4} \implies r^2 = 36 \implies r = 6$$ [cite: 1387] (Choose +6$+6$ because terms must stay strictly positive [cite: 655]).

Find value for first base variable a$a$ [cite: 1389]: a(6) = (3)/(4) a = (1)/(8)$$a(6) = \frac{3}{4} \implies a = \frac{1}{8}$$ [cite: 1389]

Step 2: Sum configuration resolving

Now compute targeted expansion expression value [cite: 1390]: 24(a₁ + a₂ + a₃) = 24(a + ar + ar²) = 24a(1 + r + r²)$$24(a_1 + a_2 + a_3) = 24(a + ar + ar^2) = 24a(1 + r + r^2)$$ [cite: 1390] = 24 · ((1)/(8)) · (1 + 6 + 36) = 3 · 43 = 129$$= 24 \cdot \left(\frac{1}{8}\right) \cdot (1 + 6 + 36) = 3 \cdot 43 = 129$$ [cite: 1390, 1391]

Pattern Recognition

Product entries like a₃ a₅ = a₄²$a_3 a_5 = a_4^2$ help identify the central term index value quickly in symmetric geometric progressions.

Chapter Mix

Class 11 Mathematics: Sequences and Series

Core Logic

The general term Tᵣ$T_r$ of the series can be written as:

Let's factorize the denominator by completing the square metric:

Using the difference of squares identity A² - B² = (A-B)(A+B)$A^2 - B^2 = (A-B)(A+B)$:

Step 1: Partial Fraction Decomposition

Express Tᵣ$T_r$ using partial fractions split:

Notice that if we define V(r) = r² - r + 2$V(r) = r^2 - r + 2$, then V(r+1) = (r+1)² - (r+1) + 2 = r² + 2r + 1 - r - 1 + 2 = r² + r + 2$V(r+1) = (r+1)^2 - (r+1) + 2 = r^2 + 2r + 1 - r - 1 + 2 = r^2 + r + 2$.

Thus, Tᵣ = 2[V(r) - V(r+1)]$T_r = 2\big[V(r) - V(r+1)\big]$, which sets up a clear telescoping sum formulation.

Step 2: Evaluating the Sum of 20 Terms

Summing from r = 1$r = 1$ to 20$20$:

All sequential middle terms cancel completely, leaving only first and final values:

Since 210$210$ and 211$211$ are coprime, m = 210$m = 210$ and n = 211$n = 211$.

Step 3: Calculating m + n

Combining both values:

Pattern Recognition

The polynomial factorization r⁴ + a²r² + b⁴$r^4 + a^2r^2 + b^4$ is a frequent pattern in series problems. Always complete the square to break it into a product of quadratic expressions, which naturally yields a telescoping sequence.

Chapter Mix

Class 11 Mathematics: Sequences and Series