Basics

Progression

Arithmetic, geometric, and harmonic progressions with their means and key relationships.

A progression is a sequence whose terms follow a fixed rule. The three classical progressions (arithmetic, geometric, harmonic) each generate a mean (AM, GM, HM) and a closed-form expression for the n-th term and partial sum, so you do not have to compute term by term. HP is the reciprocal of an AP, and for positive numbers the three means satisfy AM ≥ GM ≥ HM.

Arithmetic Progression (AP)

An arithmetic progression is a sequence in which each term after the first is obtained by adding a constant d (the common difference) to the preceding term. The generic AP is a, a + d, a + 2d, a + 3d, … where a is the first term. Plotting the terms on a number line gives equally-spaced points.

Successive terms of an AP are equally spaced; each gap equals the common difference d.

• n^th term of AP

The n-th term adds the common difference n - 1 times to the first term. The shift of one reflects that the first term contributes zero copies of d.

t_{n}=a+(n-1)d

(Where t_n = n^th term, a = first term, 1 = last term and d = common difference)

• Number of term in an AP

Solving the n-th term formula for n counts the terms between a known first and last term. Use it when the endpoints of a finite AP are visible but the length is not (for instance, counting multiples of three between two given values).

n=\frac{\left(1+a\right)}{d}+1

(Where n = number of terms, a = first term, 1 = last term and d = common difference)

• Sum of first n terms in AP

The partial sum is n times the average of the first and last terms. Gauss’s pairing trick gives the shortcut: pair the first term with the last, the second with the second-to-last, and so on. Each pair sums to a + l, and there are n/2 such pairs. The two forms below differ only in whether the last term is given directly or expressed through a and d.

S_{n}=\left(\frac{n}{2}\right)\left[2a+\left(n+1\right)d\right]

Equivalent form using the last term directly. This is the most compact way to write the partial sum when both endpoints are known:

\text{ }\text{ }\text{ }\text{ }=\left(\frac{n}{2}\right)\left[a+1\right]

(Whare a = first term, d = commn difference and 1 = n^th term = a +(n-1)d)

• Arithmtic Mean

If a, b, c are in AP, then b is the Arithmetic Mean (A.M.) of a and c. The AM is the centre value that two endpoints determine to form a three-term AP.

The middle term of three numbers in AP is the average of the outer two. The constant-difference condition b - a = c - b rearranges to:

b=\frac{a+c}{2}

A.M between two numbers a and b. Any two numbers can be treated as the outer terms of a three-term AP whose centre is their average:

\text{ }\text{ }=\frac{a+b}{2}

If a, a₁, a₂, … a_n, b are in AP we can say that a₁, a₂, … a_n are the n Arithmetic Means berween a and b.

• Nots

Most AP problems simplify when the terms are taken symmetrically around the central term. The common-difference contributions cancel pairwise on summing: for odd counts the d terms cancel directly, and for even counts they cancel in symmetric pairs of ±d, ±3d, and so on.

3\text{ }terms\text{ }: \left(a-d\right), a,\left(a+d\right)

4\text{ }terms\text{ }: \left(a-3d\right), \left(a-d\right),\left(a+d\right),\left(a+3d\right)

5\text{ }terms\text{ }: \left(a-2d\right), \left(a-d\right),a,\left(a+d\right),\left(a+2d\right)

T_n = S_n - S_n-1

Geometric Progression (GP)

A geometric progression is a sequence of non-zero numbers in which the ratio of consecutive terms is a fixed constant r. AP is built on addition of a fixed d; GP is built on multiplication by a fixed r. The parallel structure carries through to the n-th term, sum, and mean formulas below. Plotting GP terms gives exponential growth or decay.

• n^th term of GP

The n-th term multiplies the first term by the common ratio n - 1 times. The exponent shift mirrors the (n - 1)d in the AP case: the first term contributes the ratio zero times.

t_{n}=ar^{\left(n-1\right)}

(Where t_n = n^th term, a = first term, r = common ratio and n = number of terms)

• Sum of first n Terms in GP

The partial sum is a closed-form geometric series. The two branches differ only in sign arrangement so the numerator stays positive in each regime; they are algebraically equivalent. The derivation is the standard S - rS telescoping trick, which collapses all interior terms.

Sn = \left\{ \begin{array}{l l} \frac{a\left(r^{n}-1\right)}{r-1} & \quad \text{ }if\text{ }\left(r > 1\right)\\ \frac{a\left(1-r^{n}\right)}{1-r} & \quad \text{ }if\text{ }\left(r < 1\right)\ \end{array} \right.

(Where a = the first term, r = common ratio, n = number of terms)

Note: When n = ∞, then

For a convergent infinite GP (ratio strictly between 0 and 1), the series sums to a finite limit. The term rⁿ goes to zero as n grows, so the partial-sum formula collapses to the ratio shown. This is the formula behind repeating-decimal-to-fraction conversion.

S_{\infty}=\frac{a}{1-r} \text{ }\left(if\text{ } 0<r<1\right)

• Geometic Mean

If three non-zero numbers a, b, c are in GP, then b is the Geometric Mean (GM) of a and c. The GM plays the multiplicative role the AM plays additively: it is the value for which the ratios b/a and c/b match.

The geometric mean of two numbers is the square root of their product. The constant-ratio condition gives b² = ac:

b=\sqrt{ac}

The GM of two arbitrary numbers a and b follows the same template:

b=\sqrt{ab}

(Note that if a and b are of opposite sign, their GM is not defined)

• Note

If a, b, c are in GP, the following ratio identity holds. It follows from b/a = c/b:

\frac{a-b}{b-c}=\frac{a}{b}

Most GP problems simplify when the terms are taken in a symmetric multiplicative form. The central term is unaffected, and ratio terms pair as r and 1/r so products simplify.

3\text{ }terms\text{ }:\left(\frac{b}{r}\right),a,ar

5\text{ }terms\text{ }:\left(\frac{a}{r^{2}}\right),\left(\frac{a}{r}\right),a,ar,ar^{2}

Harmonic Progression (HP)

A harmonic progression is a sequence of non-zero numbers whose reciprocals form an AP. HP arises in averaging rates (speeds over a fixed distance, parallel resistances, capacitances in series) where the linear quantity is the reciprocal of what you measure directly.

A harmonic progression is defined by the reciprocals of its terms forming an AP:

\text{ }\frac{1}{a1},\frac{1}{a2},\frac{1}{a3}, ... \frac{1}{an}\text{ }

are in AP

• If a, (a+d), (a+2d), … are in AP, the n-th term of the AP is a + (n-1)d. So if

\text{ }\frac{1}{a},\frac{1}{\left(a+d\right)},\frac{1}{\left(a+2d\right)},...\text{ }

are in HP, the n-th term of the HP is the reciprocal of the corresponding AP term. HP has no independent closed form for the partial sum; unlike AP and GP, partial sums of HPs are not generally expressible in elementary closed form.

\text{ }HP=\frac{1}{\left[a+\left(n-1\right)d\right]}\text{ }

• If a, b, c are in HP, then b is the Harmonic Mean (HM) of a and c. By definition, the HM is the reciprocal of the AM of the reciprocals. For positive inputs it is always less than or equal to both the AM and the GM.

The harmonic mean is twice the product divided by the sum, from inverting 2/b = 1/a + 1/c:

\text{ }b=\frac{\left(2ac\right)}{\left(a+c\right)}\text{ }

• The HM of two arbitrary numbers follows the same template:

\text{ }b=\frac{\left(2ab\right)}{\left(a+b\right)}\text{ }

• If a, b, c are in HP, the relation reads more cleanly in reciprocals. 1/b is the average of 1/a and 1/c, which is the AP condition on the reciprocals:

\text{ }, \text{ }\frac{2}{b}=\frac{1}{a}+\frac{1}{c}

Relationship between AM, GM and HM of Two Numbers

For any two positive numbers the three means stand in a fixed relationship. The identity below links the three: knowing any two of AM, GM, HM determines the third. The chain AM ≥ GM ≥ HM (with equality only when the inputs are equal) is the two-number case of the classical inequality of means.

If GM, AM, and HM are the Geometric, Arithmetic, and Harmonic means of two positive numbers, then the GM is the geometric mean of the AM and HM. Squaring gives the symmetric form: GM² equals their product.

GM^{2}= AM \times HM