Sum of n Terms of an Arithmetic Progression :

To find the sum of the series, we use the formula given below.

S_{n} = (n/2) [a + l] (or)

S_{n} = (n/2) [2a + (n - 1)d]

To find the first term, common difference and number of term, we use the formula given below.

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

**Question 1 :**

If the sum of 7 terms of an AP is 49 and that of 17 terms is 289,find the sum of first n terms.

**Solution :**

S_{7} = 49

S_{17} = 289

S_{n} = (n/2) [2a + (n - 1) d]

S_{7} = (7/2) [2 a + (7 - 1) d]

49 = (7/2) [ 2a + 6d]

(49 x 2)/7 = 2 a + 6 d

14 = 2 a + 6 d

a + 3 d = 7 -----(1)

S_{17 }= (17/2) [2 a + (17 - 1) d]

289 = (17/2) [ 2a + 16d]

(289 x 2)/17 = 2 a + 16 d

34 = 2 a + 16 d

a + 8 d = 17 -----(1)

(1) - (2)

a + 3 d = 7

a + 8 d = 17

(-) (-) (-)

-------------

- 5 d = -10

d = 2

By applying the value of d in (1), we get

a + 3 (2) = 7

a + 6 = 7

a = 7 - 6

a = 1

To find the sum of first n terms, we have to apply the values of a and in the Sn formula

S_{n} = (n/2) [2a + (n - 1) d]

S_{n} = (n/2) [2(1) + (n - 1) (2)]

= (n/2) [2 + 2 n - 2]

= (n/2) [2 n]

= n²

**Question 2 :**

Show that a₁, a₂,............ an form an AP where an is defined as below

(i) a_{n} = 3 + 4 n

(ii) a_{n} = 9 - 5 n

Also find the sum of 15 terms in each case.

**Solution :**

(i) a n = 3 + 4 n

n = 1 a₁ = 3 + 4(1) = 7 |
n = 2 a₂ = 3 + 4(2) = 11 d = a₂ - a₁ = 11 - 7 d = 4 |

So the series will be in the form 7 + 11 + ...........

Now we need to find sum of 15 terms

S_{15} = (n/2) [ 2 a + (n - 1) d]

= (15/2) [2(7) + (15-1) 4]

= (15/2) [14 + 14(4)]

= (15/2) [14 + 56]

= (15/2) [70]

= (15 x 35)

S_{15} = 525

(ii) a_{n} = 9 - 5 n

**Solution :**

n = 1 a |
n = 2 a d = a = -1 - 4 = -5 |

So the series will be in the form 4 + (-1) + ...........

Now we need to find sum of 15 terms

S_{15} = (n/2) [ 2 a + (n - 1) d]

= (15/2) [2(4) + (15-1) (-5)]

= (15/2) [8 + 14(-5)]

= (15/2) [8 - 70]

= (15/2) [-62]

= 15 x (-31)

= -465

After having gone through the stuff given above, we hope that the students would have understood, sum of n terms of an arithmetic progression.

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