= = 1 t1 a, tn+1 rtn n r ≥ for and where is the constant ratio Partial sums of a sequence (Series) t1,…, ts, The sequence of partial sums of a sequence … is defined by = + Sn t1 tn …+ Partial sum of an arithmetic sequence (Arithmetic series) Sn n a The partial sum of the first terms of an arithmetic sequence with first term and common difference d. + + 2d,……, + (n 1)d,… a, a d, a a – is = (a + = (2a + (n 1)d) Sn tn) tn nth – where is the term of the sequence. = + = Sn+1 Sn tn S1 t1 The partial sums form a sequence with and Partial sums of a geometric sequence (Geometric series) Sn n a r, The partial sum of the first terms of a geometric sequence with first term and common ratio a, ar, ar2,……, arn – 1,… is = , 1. Sn r ≠ = + = Sn+1 Sn tn S1 t1. The partial sums form a sequence with and Introduction to differential calculus Gradient (Slope) The gradient of the straight line passing through points and is the ratio . Slope is a synonym for gradient. Secant A secant of the graph of a function is the straight line passing through two points on the graph. The line segment between the two points is called a chord. Tangent line The tangent line (or simply the tangent) to a curve at a given point can be described intuitively as the straight line that "just touches" the curve at that point. At where the tangent meets the curve the curve meet, the curve has "the same direction" as the tangent line. In this sense it is the best straight-line approximation to the curve at the point . Linearity property of the derivative The linearity property of the derivative is summarized by the equations: for any constant and Local and global maximum and minimum
t1 = a, tn+1 = rtn for n ≥ 1 and where r is the constant ratio Partial sums of a sequence (Series) The sequence of partial sums of a sequence t1,…, ts, … is defined by Sn = t1 + …+ tn Partial sum of an arithmetic sequence (Arithmetic series) The partial sum Sn of the first n terms of an arithmetic sequence with first term a and common difference d. a, a + d, a + 2d,……, a + (n – 1)d,… is Sn = (a + tn) = (2a + (n – 1)d) where tn is the nth term of the sequence. The partial sums form a sequence with Sn+1 = Sn + tn and S1 = t1 Partial sums of a geometric sequence (Geometric series) The partial sum Sn of the first n terms of a geometric sequence with first term a and common ratio r, a, ar, ar2,……, arn – 1,… is Sn = , r ≠ 1. The partial sums form a sequence with Sn+1 = Sn + tn and S1 = t1. Introduction to differential calculus Gradient (Slope) The gradient of the straight line passing through points and is the ratio . Slope is a synonym for gradient. Secant A secant of the graph of a function is the straight line passing through two points on the graph. The line segment between the two points is called a chord. Tangent line The tangent line (or simply the tangent) to a curve at a given point can be described intuitively as the straight line that "just touches" the curve at that point. At where the tangent meets the curve the curve meet, the curve has "the same direction" as the tangent line. In this sense it is the best straight-line approximation to the curve at the point . Linearity property of the derivative The linearity property of the derivative is summarized by the equations: for any constant and Local and global maximum and minimum
AC_Mathematical_Methods_T_14-20.docx