Let < aₙ >$< a_n >$ be a sequence such that a₀ = 0, a₁ = (1)/(2)$a_0 = 0, a_1 = \frac{1}{2}$ and 2an + 2 = 5an + 1 - 3aₙ, n = 0, 1, 2, 3,$2a_{n + 2} = 5a_{n + 1} - 3a_n, n = 0, 1, 2, 3, \dots$. Then Σk = 1¹⁰⁰ ak$\sum_{k = 1}^{100} a_k$ is equal to:

Related Formula

Core Logic

Given S₃ = 54 3a + 3d = 54 a + d = 18$S_3 = 54 \implies 3a + 3d = 54 \implies a + d = 18$. Express S₂₀$S_{20}$ as:

Substitute a = 18 - d$a = 18 - d$ into the expression:

Step 1: Formulate Inequality and Constraint Bound

Given 1600 < S₂₀ < 1800$1600 < S_{20} < 1800$:

124 < 17d < 144$124 < 17d < 144$

Step 2: Isolate Integer Term parameters

Since the sequence consists of positive integers, common difference d$d$ must be an integer d = 8$\implies d = 8$. Then a = 18 - 8 = 10$a = 18 - 8 = 10$.

Step 3: Calculate the 11th Term

Pattern Recognition

Diophantine properties (integer conditions) drastically restrict valid inequality windows. Always check parameters for strict divisibility to skip unnecessary computation.

Chapter Mix

Class 11 Mathematics: Sequences and Series

Related Formula

• An Arithmetic Progression (A.P.) with common difference d$d$ sets consecutive terms as: Tₙ = T₁ + (n-1)d$T_n = T_1 + (n-1)d$
• A Geometric Progression (G.P.) ensures: T₂² = T₁ · T₃$T_2^2 = T_1 \cdot T_3$

Core Logic

Since 3, a, b, c$3, a, b, c$ are elements of an A.P., let d$d$ denote the common difference:

• a = 3 + d$a = 3 + d$
• b = 3 + 2d$b = 3 + 2d$
• c = 3 + 3d$c = 3 + 3d$

Substituting these values into the sequence configurations of the given G.P. (3, a-1, b+1, c+9$3, a-1, b+1, c+9$):

Step 1: Compute the Common Difference

Using the geometric mean property for the first three terms (3, 2+d, 4+2d$3, 2+d, 4+2d$):

Step 2: Evaluate both cases for the Progressions

• Case A: If d = 4$d = 4$

The G.P. sequence reads: 3, 6, 12, 24$3, 6, 12, 24$ (common ratio r=2$r=2$, valid layout). The values are: a = 7, b = 11, c = 15$a = 7, b = 11, c = 15$.

• Case B: If d = -2$d = -2$

The G.P. sequence reads: 3, 0, 0, 6$3, 0, 0, 6$ (contains zeros, violating standard geometric definitions).

Hence, select d = 4$d = 4$.

Step 3: Calculate the Final Arithmetic Mean

The required arithmetic mean of a, b, c$a, b, c$ is:

Pattern Recognition

Sees: Transition parameters mapping from A.P. linear spacing into G.P. ratios. Shortcut: Notice that the arithmetic mean of a, b, c$a, b, c$ is exactly equal to the middle value b$b$ for any linear sequence. Thus, finding b = 3 + 2(4) = 11$b = 3 + 2(4) = 11$ directly yields the final answer.

Chapter Mix

Class 11 Mathematics: Sequences and Series

Related Formula

• General term of an AP: Tₙ = a + (n-1)d$T_n = a + (n-1)d$
• Sum of n$n$ terms of an AP: Sₙ = (n)/(2)[2a + (n-1)d]$S_n = \frac{n}{2}[2a + (n-1)d]$
• The common difference of a series of common terms between two APs is given by the least common multiple of their respective common differences:

Core Logic

Let's analyze both arithmetic progressions:

• AP 1: 3, 7, 11, 15, , 403$3, 7, 11, 15, \dots, 403 \implies$ First term a₁ = 3$a_1 = 3$, common difference d₁ = 4$d_1 = 4$.
• AP 2: 2, 5, 8, 11, , 404$2, 5, 8, 11, \dots, 404 \implies$ First term a₂ = 2$a_2 = 2$, common difference d₂ = 3$d_2 = 3$.

By observation, the first identical value appearing in both series is 11$11$. Therefore, the new common AP has:

• First term a = 11$a = 11$
• Common difference d = LCM(4, 3) = 12$d = \text{LCM}(4, 3) = 12$

Step 1: Determine the Number of Common Terms

The last term Tₙ$T_n$ of the common AP cannot exceed the boundary upper limits of either individual series (i.e., ≤ 403$\le 403$):

12(n-1) ≤ 392$12(n-1) \le 392$

Step 2: Calculate the Series Sum

Using the AP summation formula for 33$33$ terms:

Pattern Recognition

Sees: Overlapping arithmetic progression series elements. Shortcut: Once the first matching number and the LCM are calculated, the maximum term inequality a + (n-1)d ≤ (L₁, L₂)$a + (n-1)d \le \min(L_1, L_2)$ directly maps out the total number of terms cleanly.

Chapter Mix

Class 11 Mathematics: Sequences and Series

Related Formula

If x, y, z$x, y, z$ are in A.P., then 2y = x + z$2y = x + z$.

Core Logic

From the first sequence condition:

From the second sequence condition, the components are ₑ((a)/(2b)), ₑ((2b)/(3c)), ₑ((3c)/(a))$\log_e\left(\frac{a}{2b}\right), \log_e\left(\frac{2b}{3c}\right), \log_e\left(\frac{3c}{a}\right)$:

Step 1: Finding Ratios

Substituting c = (2b)/(3)$c = \frac{2b}{3}$ into equation (i):

Thus, consolidating all parts: a : b = 9 : 6$a : b = 9 : 6$ b : c = 6 : 4$b : c = 6 : 4$

Pattern Recognition

Logarithmic A.P. strings immediately translate to simple geometric proportions inside the core arguments via logarithmic properties (2 x = x²$2\log x = \log x^2$).

Chapter Mix

Class 11 Mathematics: Sequences and Series

Related Formula

Core Logic

Let the terms of the Geometric Progression have a common ratio r$r$. Substituting the geometric forms into the arithmetic mean relationship:

Dividing out non-zero fields a rⁿ⁻¹$a r^{n-1}$:

Since a₂ ≠ a₁$a_2 \neq a_1$, we have r ≠ 1$r \neq 1$. Thus, the common ratio is r = -2$r = -2$.

Step 1: Evaluating the Target Partial Difference

We need to evaluate:

Substituting a = (1)/(8)$a = \frac{1}{8}$ and r = -2$r = -2$:

Pattern Recognition

Partial sum differences simplify into standard standalone term values (Sₙ - Sₙ₋₂ = Tₙ + Tₙ₋₁$S_n - S_{n-2} = T_n + T_{n-1}$). This eliminates the need to apply long fraction sum formats.

Chapter Mix

Class 11 Mathematics: Sequences and Series