Mathematical Induction & Summation

Mathematical Induction is a proof technique for establishing that a statement P(n) holds for all natural numbers n >= n0. It is rarely tested as a standalone proof in JEE Main but is essential background for summation formulas and series manipulation.

Principle of Mathematical Induction (PMI):

1. Base Case: Verify P(n0) is true (usually n0 = 1).
2. Inductive Step: Assume P(k) is true for some arbitrary k >= n0 (Inductive Hypothesis). Prove P(k+1) is true using this assumption.
3. Conclusion: P(n) is true for all n >= n0.

Strong Induction: Assume P(m) is true for ALL m with n0 <= m <= k, then prove P(k+1). Useful when P(k+1) depends on multiple predecessors, not just P(k).

Standard Summation Formulas (must memorize):

• sum(k, k=1 to n) = n(n+1)/2
• sum($k^{2}$, k=1 to n) = n(n+1)(2n+1)/6
• sum($k^{3}$, k=1 to n) = [n(n+1)/2]² = [sum(k)]²
• sum($k^{4}$, k=1 to n) = n(n+1)(2n+1)($3n^{2}$+3n-1)/30

Telescoping Series: If a series can be written as sum of [f(k) - f(k+1)], most terms cancel, leaving f(1) - f(n+1). This is the primary technique for JEE summation problems.

Partial Fraction Decomposition for Summation: For rational terms like 1/[k(k+1)], decompose: 1/[k(k+1)] = 1/k - 1/(k+1). Sum telescopes to 1 - 1/(n+1) = n/(n+1).

Method of Differences: If a sequence {an} has the property that successive differences form a recognizable pattern (e.g., an AP or GP), use the method of differences: an = a1 + sum($d_{k}$, k=1 to n-1) where $d_{k}$ = a_{k+1} - $a_{k}$.

$\Sigma$ Notation Properties:

• sum(cf(k)) = csum(f(k))
• sum(f(k)+g(k)) = sum(f(k)) + sum(g(k))
• sum(f(k), k=m to n) = sum(f(k), k=1 to n) - sum(f(k), k=1 to m-1)

Vn Method: For sums of products of terms in AP, if $T_{r}$ = V_{r+1} - $V_{r}$ where $V_{r}$ is a product of consecutive AP terms, then sum($T_{r}$) telescopes.
Example: r(r+1) = [r(r+1)(r+2) - (r-1)r(r+1)]/3.

The key problem-solving concept is recognizing when a summation can be telescoped or simplified using partial fractions, and knowing the standard summation formulas to evaluate polynomial sums directly.