Sequences and Series — JEE Mains Mathematics

Formulas (3)

• The nth term and the common difference · nth term
  $$a_n=a+(n-1)d,\qquad d=\frac{a_q-a_p}{q-p}$$
• Sum of n terms, and an AP given by its sum · Sum of n terms
  $$S_n=\frac n2\big[2a+(n-1)d\big]$$
• Symmetric terms, equal pairs and odd–even parts · Odd–even parts of 2k terms
  $$S_{\text{even}}-S_{\text{odd}}=kd,\qquad a_k+a_{n+1-k}=a_1+a_n$$

Watch out for (3)

• Count steps, not terms
  → The nth term and the common difference
• A constant term in S_n
  → Sum of n terms, and an AP given by its sum
• Odd-placed, not odd-valued
  → Symmetric terms, equal pairs and odd–even parts

Formulas (2)

• Common terms of two progressions · Common difference of the shared terms
  $$D=\operatorname{lcm}(d_1,d_2)$$
• Terms picked out by a divisibility condition · Sum of the terms that pass
  $$\text{sum}=S_{\text{all}}-S_{\text{excluded}}$$

Watch out for (2)

• Start at the first COMMON term
  → Common terms of two progressions
• Add the overlap back
  → Terms picked out by a divisibility condition

Formulas (3)

• Terms and sums of a GP from two conditions · Sum of n terms
  $$S_n=a\,\frac{r^n-1}{r-1}$$
• Products, means and odd–even parts of a GP · Whole sum of 2m terms
  $$S=(1+r)\,S_{\text{odd}}$$
• Recognising a GP in a function, a recurrence or a two-base sum · Two-base sum
  $$\sum_{k=0}^{m}x^{k}y^{m-k}=\frac{y^{m+1}-x^{m+1}}{y-x}$$

Watch out for (3)

• Increasing and positive means r > 1
  → Terms and sums of a GP from two conditions
• An odd number of terms
  → Products, means and odd–even parts of a GP
• f(x + y) = 2f(x)f(y) is not f(1)^n
  → Recognising a GP in a function, a recurrence or a two-base sum

Formulas (3)

• Turning 'in AP', 'in GP' and 'in HP' into equations · Three conditions
  $$2b=a+c,\qquad b^2=ac,\qquad b=\frac{2ac}{a+c}$$
• AM, GM and HM of two numbers, and means inserted between them · The three means
  $$A\ge G\ge H,\qquad G^2=A\,H$$
• Least and greatest values by AM–GM · AM–GM
  $$\frac{x_1+x_2+\dots+x_n}{n}\ge\left(x_1x_2\cdots x_n\right)^{1/n}$$

Watch out for (3)

• A zero term is not a GP
  → Turning 'in AP', 'in GP' and 'in HP' into equations
• The larger root is the AM
  → AM, GM and HM of two numbers, and means inserted between them
• Check the equality case
  → Least and greatest values by AM–GM

Formulas (2)

• Sum to infinity, and the series of squares · Sum to infinity
  $$S_\infty=\frac{a}{1-r},\quad |r|<1$$
• Infinite GPs inside exponents, logarithms and figures · The geometric series
  $$1+x+x^2+\dots=\frac{1}{1-x},\quad |x|<1$$

Watch out for (2)

• Check |r| < 1
  → Sum to infinity, and the series of squares
• The ratio must stay below 1
  → Infinite GPs inside exponents, logarithms and figures

Formulas (3)

• Σk, Σk² and Σk³ applied to a polynomial kth term · Sum of cubes
  $$\sum_{k=1}^{n}k^3=\left[\frac{n(n+1)}{2}\right]^2$$
• Finding the kth term from differences or from the sum · Quadratic kth term
  $$T_k=ak^2+bk+c,\qquad 2a=\text{second difference}$$
• Alternating, grouped and greatest-integer sums · Alternating squares
  $$1^2-2^2+3^2-\dots-(2n)^2=-n(2n+1)$$

Watch out for (3)

• Sum first, then put in the limit
  → Σk, Σk² and Σk³ applied to a polynomial kth term
• Check the fourth term
  → Finding the kth term from differences or from the sum
• An odd count leaves one term unpaired
  → Alternating, grouped and greatest-integer sums

Formulas (3)

• Partial fractions that cancel in pairs · Telescoping
  $$\sum_{k=1}^{n}\big[f(k)-f(k+1)\big]=f(1)-f(n+1)$$
• Quartic denominators and surds · Quartic split
  $$\frac{k}{k^4+k^2+1}=\frac12\left[\frac{1}{k^2-k+1}-\frac{1}{k^2+k+1}\right]$$
• Factorials, powers and functions that telescope · Factorial telescoping
  $$k\cdot k!=(k+1)!-k!$$

Watch out for (3)

• A gap of two leaves two terms at each end
  → Partial fractions that cancel in pairs
• Keep the ½
  → Quartic denominators and surds
• The leftover at the lower limit
  → Factorials, powers and functions that telescope

Formulas (3)

• Multiply by the ratio and subtract · Infinite AGP
  $$S_\infty=\frac{a}{1-r}+\frac{dr}{(1-r)^2}$$
• Numerators whose differences are in AP · Quadratic numerators
  $$\sum_{k\ge1}k(k+1)\,x^{k-1}=\frac{2}{(1-x)^3}$$
• Series summed through e and the logarithm series · The exponential pieces
  $$\sum_{n\ge0}\frac{n(n-1)}{n!}=\sum_{n\ge0}\frac{n}{n!}=e$$

Watch out for (3)

• The finite case has a last term
  → Multiply by the ratio and subtract
• One subtraction is not enough
  → Numerators whose differences are in AP
• Watch where the sum starts
  → Series summed through e and the logarithm series