Question

find the sum of arethametic progress 34+32+30+......+10​

Answer 1

Given ,

First term (a) = 34

Common difference (d) = -2

Last term (l) = 10

We know that , the nth term of an AP is given by

$\boxed{ \tt{ a_{n} = a + (n - 1)d}}$

Thus ,

10 = 34 + (n - 1)(-2)

-24 = (n - 1)(-2)

12 = n - 1

n = 13

Now , the sum of first n terms of an AP is given by

$\boxed{ \tt{s = \frac{n}{2}(a + l) }}$

Thus ,

s = 13/2 × (34 + 10)

s = 13/2 × 44

s = 13 × 22

s = 286

Therefore , the sum of given arithmetic series is 286

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Answer 2

Answer:

AP  ( Arithmetic progression).

 

A list of numbers a1, a2, a3  ………….. an is called an arithmetic progression ,

Is there exists a constant number ‘d’

a2= a1+d

a3= a2+d

a4= a3+d……..

an= an-1+d ………

Each of the numbers in the list is called a term .

An arithmetic progression is a list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term.

This fixed  number is called the common difference( d ) of the AP.

General form of an AP.:

a, a+d, a+2d, a+3d…….

Here a is the first term and d is common difference.

Sum of n terms of an AP

The sum of first n terms of an AP with first term 'a' and common difference 'd' is given by

Sn = n /2 [ 2a + ( n - 1) d] or 

Sn=n /2 [ a + l ]       (l = last term)

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Solution:

(i)Given:

a = 7

l = an= 84

d = a2 − a1 =   – 7 = 21/2 – 7 = 7/2

d= 7/2

Let 84 be the nth term of this A.P.

l = a (n – 1)d

84 = 7 + (n – 1) × 7/2

77 = (n – 1) × 7/2

22 = n − 1

n = 23

We know that,

Sn = n/2 (a + l)

Sn = 23/2 (7 + 84)

      = (23×91/2) = 2093/2

 Sn=  1046 1/2

(ii) 34 + 32 + 30 + ……….. + 10

Given:

a = 34

d = a2 − a1 = 32 − 34 = −2d= -2l = an= 10

Let 10 be the nth term of this A.P.

l = a + (n − 1) d

10 = 34 + (n − 1) (−2)

−24 = (n − 1) (−2)

12 = n − 1

n = 13

Sn = n/2 (a + l)

= 13/2 (34 + 10)

= (13×44/2) = 13 × 22

Sn= 286

(iii) (−5) + (−8) + (−11) + ………… + (−230)

Given:

a = −5

l = −230

d = a2 − a1 = (−8) − (−5)

= − 8 + 5 = −3

d= -3

Let −230 be the nth term of this A.P.

l = a + (n − 1)d

−230 = − 5 + (n − 1) (−3)

−225 = (n − 1) (−3)

(n − 1) = 75

n = 76

Sn = n/2 (a + l)

= 76/2 [(-5) + (-230)] = 38×(-235)

Sn= -8930

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Hope this will help you....