Question

The sum of the first eight terms of an arithmetic series is 180 and it's fifth term is tive times of the first term. Find the first term and the common difference.
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Answer 1

Step-by-step explanation:

Given :-

The sum of the first eight terms of an arithmetic series is 180.

It's fifth term is five times of the first term.

To find :-

i) The first term

ii) The common difference.

Solution :-

Let the first term of the A.P. be a

Let the common difference be d

We know that

nth term of an A.P. = a+(n-1)d

Fifth term of the A.P. = a+(5-1)d = a+4d

Given that

Fifth term of the A.P. = 5×First term

=> a+4d = 5a

=> 4d = 5a-a

=> 4d = 4a

=> d = 4a/4

=> d = a ------------(1)

Therefore, First term = Common difference

We know that

Sum of the first 'n' terms of an AP

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

We have,

Sum of the first 8 terms of the A.P. = 180

n = 8

=> (8/2)[2a+(8-1)d] = 180

=> 4(2a+7d) = 180

=>4(2a+7a) = 180 (from (1))

=> 4(9a) = 180

=> 36a = 180

=> a = 180/36

=> a = 5

Therefore, a = 5

Therefore, d = 5

Answer :

I) First term of the A.P. = 5

ii) Common difference of the A.P. = 5

Check:-

We have, a = 5, d = 5

Fifth term = a+4d

= 5+4(5)

= 5+20

= 25

and

5 times a = 5(5) = 25

Therefore,

Fifth term = 5 times the first term

and

Sum of the first 8 terms of the A. P.

= (8/2)[2(5)+(8-1)(5)]

= 4(10+35)

= 4(45)

= 180

Verified the given relations in the given problem.

Used formulae:-

→ nth term of an A.P. = a+(n-1)d

→ Sum of the first 'n' terms of an AP

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

• a = First term
• d = Common difference
• n = Number of terms

Answer 2

Step-by-step explanation:

Step-by-step explanation:

Given :-

The sum of the first eight terms of an arithmetic series is 180.

It's fifth term is five times of the first term.

To find :-

i) The first term

ii) The common difference.

Solution :-

Let the first term of the A.P. be a

Let the common difference be d

We know that

nth term of an A.P. = a+(n-1)d

Fifth term of the A.P. = a+(5-1)d = a+4d

Given that

Fifth term of the A.P. = 5×First term

=> a+4d = 5a

=> 4d = 5a-a

=> 4d = 4a

=> d = 4a/4

=> d = a ------------(1)

Therefore, First term = Common difference

We know that

Sum of the first 'n' terms of an AP

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

We have,

Sum of the first 8 terms of the A.P. = 180

n = 8

=> (8/2)[2a+(8-1)d] = 180

=> 4(2a+7d) = 180

=>4(2a+7a) = 180 (from (1))

=> 4(9a) = 180

=> 36a = 180

=> a = 180/36

=> a = 5

Therefore, a = 5

Therefore, d = 5

Answer :

I) First term of the A.P. = 5

ii) Common difference of the A.P. = 5

Check:-

We have, a = 5, d = 5

Fifth term = a+4d

= 5+4(5)

= 5+20

= 25

and

5 times a = 5(5) = 25

Therefore,

Fifth term = 5 times the first term

and

Sum of the first 8 terms of the A. P.

= (8/2)[2(5)+(8-1)(5)]

= 4(10+35)

= 4(45)

= 180

Verified the given relations in the given problem.

Used formulae:-

→ nth term of an A.P. = a+(n-1)d

→ Sum of the first 'n' terms of an AP

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

a = First term

d = Common difference

n = Number of terms

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