For positive integers n, if 4aₙ=(n²+5n+6)$4a_{n}=(n^{2}+5n+6)$ and Sₙ=Σk=1ⁿ( 1ak)$S_{n}=\sum_{k=1}^{n}\left(\frac{1}{a_{k}}\right)$ then the value of 507 S₂₀₂₅$S_{2025}$ is:

Related Formula

Sum of first r$r$ odd natural numbers:

Exponential series expansion:

Core Logic

Let's find the general r$r$-th term of the series:

Step 1: Expressing term in terms of sum limits

Let's write r = (r-1) + 1$r = (r-1) + 1$:

Our infinite sum is:

Both sums are standard representations of the exponential expansion.

Step 2: Summing the parts

• First part: Σr=2∞ (1)/((r-2)!) = 1 + (1)/(1!) + (1)/(2!) + = e$\sum_{r=2}^{\infty} \frac{1}{(r-2)!} = 1 + \frac{1}{1!} + \frac{1}{2!} + \dots = e$
• Second part: Σr=1∞ (1)/((r-1)!) = 1 + (1)/(1!) + (1)/(2!) + = e$\sum_{r=1}^{\infty} \frac{1}{(r-1)!} = 1 + \frac{1}{1!} + \frac{1}{2!} + \dots = e$

Pattern Recognition

The general term containing r²$r^2$ in summation with factorials converges to 2e$2e$. Remember the shortcut: Σ (r²)/(r!) = 2e$\sum \frac{r^2}{r!} = 2e$, Σ (r³)/(r!) = 5e$\sum \frac{r^3}{r!} = 5e$. It is extremely useful to memorize these common limits.

Chapter Mix

Class 11 Mathematics: Sequences and Series Class 12 Mathematics: Limits, Continuity and Differentiability

Related Formula

For a geometric progression, the terms can be set as a, ar, ar², ar³$a, ar, ar^2, ar^3$. For three terms A, B, C$A, B, C$ to be in arithmetic progression, they must satisfy: 2B = A + C$2B = A + C$

Core Logic

Let the elements be x₁ = a, x₂ = ar, x₃ = ar², x₄ = ar³$x_1 = a, x_2 = ar, x_3 = ar^2, x_4 = ar^3$. After the specified subtractions, the sequence becomes:

Since this sequence is in AP, we form two separate common difference linear linkages:

Step 1: Solve the Simultaneous Polynomials

Multiply equation (1) by r$r$:

Subtract equation (3) from equation (2):

Substitute r = 2$r = 2$ back into equation (1):

Step 2: Find the Continuous Product Value

The continuous product term is:

Now divide by 24 as required:

Pattern Recognition

Notice that multiplying the first AP condition equation by r$r$ perfectly mimics the structure of the second condition equation except for the absolute scalar value, allowing direct elimination of all polynomial variable indices simultaneously.

Chapter Mix

Class 11 Mathematics: Sequences and Series

Related Formula

Core Logic

Factor out common fractions from the even terms component (β$\beta$) to represent it as a scalar multiple of the universal sum sequence.

Step 1: Simplify the Even Terms Series

Step 2: Express Alpha by Remainder Deduction

Since total sum equals α + β$\alpha + \beta$:

Step 3: Compute the Relative Ratio

(α)/(β) = ((15)/(16) ( (π⁴)/(90) ))/((1)/(16) ( (π⁴)/(90) )) = 15$\frac{\alpha}{\beta} = \frac{\frac{15}{16} \left( \frac{\pi^4}{90} \right)}{\frac{1}{16} \left( \frac{\pi^4}{90} \right)} = 15$

Pattern Recognition

For alternating p-series powers like Σ n-p$\sum n^{-p}$, the even component fractions always condense via factor steps to 2-p · Stotal$2^{-p} \cdot S_{\text{total}}$, decoupling power values cleanly from final simple scalar quotients.

Chapter Mix

Class 11 Mathematics: Sequences and Series

Related Formula

Sum of an infinite geometric progression with |r| < 1$|r| < 1$:

Core Logic

From the definition of the relation:

Since x in S = 0, 1, 2, 3,$x \in S = \{0, 1, 2, 3, \dots\}$, the output values of y$y$ represent elements of the range.

Step 1: Compute Infinite Sum

Generating elements by plugging in values of x$x$: For x = 0 y = 1$x = 0 \implies y = 1$ For x = 1 y = (2)/(5)$x = 1 \implies y = \frac{2}{5}$ For x = 2 y = ((2)/(5))²$x = 2 \implies y = \left(\frac{2}{5}\right)^2$

Sum of elements in the range:

Pattern Recognition

Convert log equations into standard exponential equations right away. A variable index belonging to whole numbers indicates an infinite GP summation scenario.

Chapter Mix

Class 11 Mathematics: Sequences and Series Class 11 Mathematics: Relations and Functions

Related Formula

Symmetry identity rule inside Arithmetic Progressions:

Core Logic

Group matching paired steps equidistant from sequence boundary ends:

The sequence of inner indices follows an AP tracking loop:

Calculate internal block element count N$N$:

Step 1: Simplify Expression Equations

Since the inner sequence contains 404 terms, they form exactly 202 symmetrical pairs. Adding a₁$a_1$ and a₂₀₂₄$a_{2024}$ introduces one more pair, resulting in 203 identical sum blocks:

Step 2: Evaluate the Total Sum

Using the standard AP sum formula:

Pattern Recognition

Progressions possess natural positional balance. Grouping symmetrical index pairs (ak + an-k+1$a_k + a_{n-k+1}$) allows factoring out variable steps right away.

Chapter Mix

Class 11 Mathematics: Sequences and Series