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Kisachek [45]
3 years ago
5

Find the sum of first seventeen numbers of an ap whose 4th and 9th tearm are -15 and -30

Mathematics
1 answer:
Radda [10]3 years ago
4 0

Answer:

The sum of first seventeen terms is - 510

Step-by-step explanation:

Given as :

The 4th term of an A.P =  t_4 =  - 15

The 9th term of an A.P =  t_9 =  - 30

For an arithmetic progression

The nth term is given as t_n = a + ( n - 1)×d

Where a is the first term and d is the common difference between numbers

<u>So, For 4th term </u>

t_n = a + ( n - 1)×d

Or, t_4 = a + ( n - 1)×d

- 15 = a + ( 4 - 1)×d

Or, - 15 = a + 3 d            .........1

<u>So, For 9th term </u>

t_n = a + ( n - 1)×d

t_9 = a + ( n - 1)×d

- 30 = a + ( 9 - 1)×d

Or, - 30 = a + 8 d            .........2

Solve eq 1 and 2

( a + 8 d ) - ( a + 3 d ) = - 30 - ( - 15)

or, ( a - a ) + ( 8 d - 3 d ) = - 30 + 15

or, 0 + 5 d = - 15

∴ d = - \frac{15}{5} = - 3

Now, put the value of d in eq 1

I.e - 15 = a + 3 × ( - 3)  

Or. - 15 = a - 9

∴ a = -15 + 9 = - 6

Now The sum of nth term is written as :

s_n = \frac{n}{2} × [ 2 × a + ( n - 1 )×d ]

Where n is the nth term

a is the first term

d is the common difference

So<u> For n = 17th term </u>

s_17 = \frac{17}{2} × [ 2 × ( - 6) + ( 17 - 1 )×( - 3) ]

Or, s_17 = \frac{17}{2} × [ - 12 - 48 ]

Or, s_17 = \frac{17}{2} × ( - 60 )

Or, s_17 = 17 × ( - 30)

∴  s_17 = - 510

Hence The sum of first seventeen terms is - 510  Answer

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