ARITHMETIC PROGRESSION PROBLEM

There is always a general formula governing some solving. The general formula of an **Arithmetic Progression** is a, a + d, a + 2d, a + 3d and so on. Thus nth term of an AP series is T_{n} = a + (n – 1) d, where T_{n} = n^{th} term and a = first term. … The sum of n terms is also equal to the **formula** where l is the last term.

1. What is the sum of 20 terms of 31/4 + 61/4 + 81/4 + …

Solution

**This is the formula used in solving the sum of an Arithmetic Progression. **Sn = n/2 [ (2n + (n – 1) ]d

**Sn = n/2 [ (2n + (n – 1) ]d**

**Sn = 20/2 [ 2 x 15/4 + (20 – 1)] 15/2**

**Sn = 10 [ 15/2 + (19)] 15/2**

**Sn = 10 (15/2 + 95/2)**

**Sn = 10 (15 + 95)/2**

**Sn = 10 x 110/2**

**Sn = 10 x 55**

**Sn = 550.**

2. Eight wood poles are to be used for pillars and the length of the poles form an Arithmetic progression (A.p) If the second pole is 2m and the sixth term is 5m, give the length of the poles.

Solution

**This is the formula used in solving the nth term of an Arithmetic Progression Tn = a + (n – 1)d**

**Tn = a + (n – 1)d**

**T2 = a + ( 2 – 1)d = 2**

**T2 = a + (1)d = 2**

**T2 = a + d = 2 equation 1**

**T6 = a + ( n – 1)d = 5**

**T6 = a + ( 6 – 1)d = 5**

**T6 = a + ( 5)d = 5**

**T6 = a + 5d = 5 equation 2**

**therefore group them i.e**

**T2 = a + d = 2 equation 1**

**T6 = a + 5d = 5 equation 2**

**subtract equation eq 1 from eq 2**

**4d = 3**

**divide both side by 4**

**4d/4 = 3/4**

**d = 0.75**

**substitute fo a using eq. 1**

**a + d = 2**

**a + 0.75 = 2**

**a = 2 – 0.75**

**a = 1.25**

**now we have gotten out first time a = 1.25 and common difference d = 0.75 so we want to find the Eight wooden poles**

**Tn = a + (n – 1)d**

**T1 = 1.25 + ( 1 -1)0.75**

**T1 = 1.25 + ( 0)0.75**

**T1 = 1.25 + 0**

**T1 = 1.25m**

**Second term**

**T2 = 1.25 + ( 2 -1)0.75**

**T2 = 1.25 + ( 1)0.75**

**T2 = 1.25 + 0.75**

**T2 = 2m**

**Third term**

**T3 = 1.25 + ( 3 -1)0.75**

**T3 = 1.25 + ( 2)0.75**

**T3 = 1.25 + 1.5**

**T3 = 2.75m**

**fourth term**

**T4 = 1.25 + ( 4 -1)0.75**

**T4 = 1.25 + ( 3)0.75**

**T4 = 1.25 + 2.25**

**T4 = 3.5m**

**Fifth term**

**T5 = 1.25 + ( 5 -1)0.75**

**T5 = 1.25 + ( 4 )0.75**

**T5 = 1.25 + 3**

**T5 = 4 . 25m**

**Sixth term**

**T6 = 1.25 + ( 6 -1)0.75**

**T6 = 1.25 + ( 5)0.75**

**T6 = 1.25 + 3.75**

**T6 = 5m**

**Seventh term**

**T7 = 1.25 + ( 7 -1)0.75**

**T7 = 1.25 + ( 6)0.75**

**T7 = 1.25 + 4.5**

**T7 = 5.75**

**Eight Term**

**T8 = 1.25 + ( 8-1)0.75**

**T8 = 1.25 + ( 7)0.75**

**T8 = 1.25 + ( 7)0.75**

**T8 = 1.25 + 5.25**

**T8 = 6.5m**

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