Important 🔖Sequence & Series 🔢
► An arithmetic progression (AP) is a sequence in which the difference between consecutive terms remains constant. The general term or nth term of an AP is given by the formula a + (n - 1)d, where a is the first term and d is the common difference.
► The sum of the first n terms of an arithmetic progression can be calculated using two formulas. It is given by (n / 2) * [2a + (n - 1)d] or alternatively by (n / 2) * [a + l], where l represents the last term of the progression.
► When solving problems that involve an arithmetic progression, choosing terms symmetrically simplifies algebraic calculations. For three terms in an AP, it is ideal to select them as (a - d), a, and (a + d), while for four terms, the selection should be (a - 3d), (a - d), (a + d), and (a + 3d).
► If every term of an arithmetic progression is increased, decreased, multiplied, or divided by a non-zero constant, the resulting sequence remains an arithmetic progression. This property demonstrates that the sequence properties adapt uniformly without breaking the arithmetic nature.
► A geometric progression (GP) is a sequence where the ratio of any term to its preceding term remains constant. The general term or nth term of a GP is defined as a * r^(n - 1), where a is the first term and r is the common ratio.
► The sum of the first n terms of a geometric progression is given by the formula a * (r^n - 1) / (r - 1) provided that the common ratio r is not equal to one. For an infinite geometric progression, the sum converges to a / (1 - r), which is valid only under the condition that the absolute value of r is strictly less than one.
► Symmetrical selection of terms is also useful for simplifying product-based geometric progression problems. For three terms in a GP, it is best to choose them as a/r, a, and ar, whereas for four terms, the ideal selection is a/r^3, a/r, ar, and ar^3.
► The arithmetic mean (AM), geometric mean (GM), and harmonic mean (HM) are single mean values inserted between two positive numbers an and b. Their respective formulas are (a + b) / 2 for the AM, the square root of a * b for the GM, and 2ab / (a + b) for the HM.
► When multiple means are inserted between two positive numbers a and b, specific algebraic properties emerge. The total sum of n arithmetic means inserted between them equals n times their single arithmetic mean, while the total product of n geometric means equals the nth power of their single geometric mean.
► The AM-GM-HM inequality states that for any set of positive real numbers, the arithmetic mean is always greater than or equal to the geometric mean, which in turn is greater than or equal to the harmonic mean. Strict equality holds if and only if all the numbers in the set are equal, and this principle is a core tool in JEE for finding the maximum or minimum values of expressions.
► An arithmetico-geometric progression (AGP) is a specialized series formed by multiplying the corresponding terms of an arithmetic progression and a geometric progression. The standard form is expressed as a, (a + d)r, (a + 2d)r^2, (a + 3d)r^3, and so on.
► To find the sum of an arithmetico-geometric progression, a specific algorithmic method is required. The entire series is multiplied by the common ratio r, the resulting series is shifted forward by one term, and it is then subtracted from the original series to reduce it into a standard geometric progression.
► The sum of an infinite arithmetico-geometric progression can be evaluated directly through a derived formula. It is given by [a / (1 - r)] + [d * r / (1 - r)^2], which holds true only when the absolute value of the common ratio r is strictly less than one.
► Sigma notation possesses linear algebraic properties that help break down complex series summation. The summation of a constant from 1 to n equals the constant multiplied by n, the summation can be distributed across separate terms, and any constant multiplier can be pulled outside the summation operator.
