The ratio of nth term of two A.P.s is 14n − 6 8n 23 then the ratio of their sum of first m terms is

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Example 7 The income of a person is Rs. 3,00,000, in the first year and he receives an increase of Rs.10,000 to his income per year for the next 19 years. Find the total amount, he received in 20 years. Income of the person in 1st year = Rs 3,00,000 Income of the person in 2nd year = Rs 3,00,000 + 10,000 = Rs 3,10,000 Income of the person in 3rd year = Rs 3,10,000 + 10,000 = Rs 3,20,000 Thus, Income received every year is 300000, 310000, 320000, This is an A.P as difference between consecutive terms is constant. 300000, 310000, 320000, Here first term = a = 300000 common difference = d = 310000 300000 = 10000 To find the total amount received in 20 years, we use the formula Sn = n/2 ( 2a + (n 1)d ) Where, Sn = sum of n terms of A.P. n = number of terms a = first term and d = common difference Here, Sn = Amount received in 20 years, n = 20 , a = 300000, d = 10000 Putting the values S20= 20/2 [2 300000 + (20 1)10000] S20= 20/2 [2 300000 + (20 1)10000] = 10 [ 600000 + 19 10000] = 10 [600000 + 190000] = 10 (790000) = 79,00,000 Thus , the total amount received at the end of 20 years is Rs. 79,00,000.

If a series is arithmetic the sum of the first n terms, denoted S n , there are ways to find its sum without actually adding all of the terms.

To find the sum of the first n terms of an arithmetic series use the formula, n terms of an arithmetic sequence use the formula,
S n = n ( a 1   +   a n ) 2 ,
where n is the number of terms, a 1 is the first term and a n is the last term.

The series 3 + 6 + 9 + 12 + ⋯ + 30 can be expressed as sigma notation ∑ n = 1 10 3 n . This expression is read as the sum of 3 n as n goes from 1 to 10

Example 1:

Find the sum of the first 20 terms of the arithmetic series if a 1 = 5 and a 20 = 62 .

S 20 = 20 ( 5   +   62 ) 2 S 20 = 670

Example 2:

Find the sum of the first 40 terms of the arithmetic sequence
2 , 5 , 8 , 11 , 14 , ⋯

First find the 40 th term:

a 40 = a 1 + ( n − 1 ) d                 = 2 + 39 ( 3 ) = 119

Then find the sum:

S n = n ( a 1 + a n ) 2 S 40 = 40 ( 2   +   119 ) 2 = 2420

Example 3:

Find the sum:

∑ k = 1 50 ( 3 k + 2 )

First find a 1 and a 50 :

a 1 = 3 ( 1 ) + 2 = 5 a 20 = 3 ( 50 ) + 2 = 152

Then find the sum:

S k = k ( a 1   +   a k ) 2 S 50 = 50 ( 5   +   152 ) 2 = 3925

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The ratio of the sum of n terms of two A.P's is 7n+1:4n+27 . Find the ratio of mth terms.