Sequences & Series (AP, GP, Special Series)

Sequences and Series is a high-application chapter testing pattern recognition, algebraic manipulation, and summation techniques. An arithmetic progression (AP) has constant common difference d: a, a+d, a+2d, .... A geometric progression (GP) has constant common ratio r: a, ar, $ar^{2}$, .... These two progressions, along with special series, form the backbone of 2-3 JEE Main questions every year.

For an AP: the nth term is $a_{n}$ = a + (n-1)d, and the sum of n terms is $S_{n}$ = n/2 * [2a + (n-1)d] = n/2 * (a + l) where l is the last term. The arithmetic mean (AM) of a and b is (a+b)/2.
Three numbers are in AP iff 2b = a + c (the middle term equals the average of the extremes).
If $S_{n}$ is given as a quadratic in n ($S_{n}$ = $An^{2}$ + Bn), then the sequence is an AP with d = 2A.

For a GP: the nth term is $a_{n}$ = ar^(n-1), and the sum of n terms is $S_{n}$ = a(r^n - 1)/(r - 1) for r != 1.
The sum to infinity of a convergent GP (|r| < 1) is $S_{inf}$ = a/(1-r). The geometric mean (GM) of a and b is $\sqrt{ab}$.
Three numbers are in GP iff $b^{2}$ = ac. The product of n terms in GP has a closed form using the middle term.

Arithmetic-Geometric Progression (AGP): A series where each term is the product of corresponding AP and GP terms: sum = a*b + (a+d)br + (a+2d)$br^{2}$ + .... The standard method is S - rS (multiply by r and subtract), which telescopes the AP part and leaves a GP to sum.

Special Series:

• Sum of first n natural numbers: n(n+1)/2
• Sum of squares: n(n+1)(2n+1)/6
• Sum of cubes: [n(n+1)/2]² = (sum of n)²
• Telescoping series: sum of f(k) - f(k+1) = f(1) - f(n+1)

Harmonic Progression (HP): a, b, c are in HP iff 1/a, 1/b, 1/c are in AP. The harmonic mean of a and b is 2ab/(a+b). AM >= GM >= HM for positive reals (with equality iff a = b).

The key problem-solving concept is identifying the type of progression (AP/GP/AGP/HP), selecting the appropriate formula, and for non-standard series, decomposing into known summable components using partial fractions or telescoping.

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