Sequence & Series Notes PDF
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Uploaded by ElatedChrysoprase1998
St. John's Public School
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These notes cover sequences and series, including arithmetic progressions (A.P.) and geometric progressions (G.P.). Key formulas and properties are outlined for both, along with examples. The notes also introduce sigma and pi notation.
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# Sequence & Series - ## A.P - * **T<sub>n</sub> = a + (n-1)d** * **S<sub>n</sub> = (2a + (n-1)d)/2** ### Properties of A.P * **Eg - 2K, 5K, 8K, 11K** * a = 2k * d = 3K * **Eg - 2/K, 5/K, 8/K, 11/K** * a = 2/k * d = 3/K * **Eg - 2+K, 5+K, 8+K, 11+K** * a = 2 + k * d = 3...
# Sequence & Series - ## A.P - * **T<sub>n</sub> = a + (n-1)d** * **S<sub>n</sub> = (2a + (n-1)d)/2** ### Properties of A.P * **Eg - 2K, 5K, 8K, 11K** * a = 2k * d = 3K * **Eg - 2/K, 5/K, 8/K, 11/K** * a = 2/k * d = 3/K * **Eg - 2+K, 5+K, 8+K, 11+K** * a = 2 + k * d = 3 * **Eg - 2-K, 5-K, 8-K, 11-K** * a = 2 - k * d = 3 * **If 3 No's are in A.P** * a, a+d, a+2d * or * a-d, a, a+d * **If 4 No's are in A.P** * a-3d, a-d, a+d, a+3d * **If 5 No's are in A.P** * a-2d, a-d, a, a+d, a+2d * **In an A.P**: * **t<sub>n</sub> - t<sub>n-1</sub> = d** (common difference) * Last term - Second last term * **In any Sequence**: * **t<sub>n</sub> - t<sub>n-1</sub> = a = constant** * **If in any sequence**: * **S<sub>n</sub> = an<sup>2</sup> + bn** (Quadratic expression in 'n' with no constant term) * **Inserting n Arithmetic Mean b/w 2 numbers**: * **A<sub>n</sub> = a + nd ; a, A<sub>1</sub>, A<sub>2</sub>, ....., A<sub>n</sub> ** * **d = (b+a)/(n-1)** *** * **Sum of all A.M blw a & b** * **Sam = n(a+b)/2 ; n -> A.M of a & b** *** ## G.P - * **t<sub>n</sub> = ar<sup>n-1</sup> ; a≠0, r≠0** * **Sum of terms in G.P** * **S<sub>n</sub> = a(1-r<sup>n</sup>)/(1-r)** * **S<sub>n</sub> = na, if r = 1** * **S<sub>0</sub> = a/(1-r) ; -1 < r < 1** * **r ≠ 0, a ≠ 0** ### Properties of G.P - * **Ka, Kar, Kar<sup>2</sup>, ..... → a.P** * a = Kar * **a, (ar), (ar<sup>2</sup>),... → a.P** * r = r = r<sup>2</sup> ### Assumption of terms - * **3 numbers in G.P** * a, ar, ar<sup>2 </sup> * or * a, a, ar * **4 No. in G.P** * a/r, a, ar, ar<sup>3</sup> * **5 No. in G.P** * a/r<sup>2</sup>, a/r, a, ar, ar<sup>2</sup> * **In a G.P / T<sub>n</sub> = r T<sub>n-1</sub>** * **Sn - Sn-1 = Tn** * T<sub>n</sub> = r T<sub>n-1</sub> * **If a, b, c are in G.P b<sup> 2 </sup> = ac** * a/b = b/c *** ## Geometric mean * **Am of a & b = (ab)<sup>1/2</sup>** * **Am of a, b & c = (abc)<sup>1/3</sup>** * **Am of a, b, c,...a<sub>n</sub> = (a, b, c,..., a<sub>n</sub>)<sup>1/n</sup>** *** ## Inserting n AM's b/w 2 +ve no. * **r =( b/a)<sup>1/n</sup>** *** ## Inequality solving A.M & G.M * **A.M ≥ a.M** *** ## Sigma Notation - 1. **Σ<sub>r=1</sub><sup>n</sup> r = n(n+1)/2** 2. **Σ<sub>r=1</sub><sup>n</sup> r<sup>2</sup> = n(n+1)(2n+1)/6** 3. **Σ<sub>r=1</sub><sup>n</sup> r<sup>3</sup> = (n(n+1)/2) <sup>2</sup>** 4. **Σ<sub>r=1</sub><sup>n</sup> r<sup>4</sup> = n(n+1)(2n+1)(3n<sup>2</sup>+3n -1)/30** 5. **Σ<sub>r=1</sub><sup>n</sup>(ar+br) = Σar + Σbr ** 6. **Σ<sub>r=1</sub><sup>n</sup>(ar-br) = Σar - Σbr ** 7. **Σ<sub>r=1</sub><sup>n </sup>k (ar) = k Σ<sub>r=1</sub><sup>n</sup> ar** 8. **Σ<sub>r=1</sub><sup>n</sup> k = nK** *** ## H.P- * a + a/1+d + a/1+2d +..... + a/1+ (n-1)d * 1/b = 1/a + 1/c = a/ac; (b = 2ac / a+c ) * **If a, b, c are in H.P mean = b = 2ac/a+c** *** ## Π - Notation - * **Π<sub>r=1</sub><sup>10</sup> (r) = 1.2.3......10** * **Π<sub>r=1</sub><sup>n</sup> (2n-1) = (1)(3)(5)... (2n-1)**
