Question

In an Ap of first term is 2 the sum of the first 5 terms is one fourth of the sum of next 5 terms find the Sum of first 20 terms

Answer 1

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ANS :

____________ -1100 ____________

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=> FOR UNDERSTANDING THESE QUESTIONS U FIRSTLY NEED TO UNDERSTAND THE FOLLOWING CONCEPTS :

=> T 1 = FIRST TERM

=> T 2 = SECOND TERM

=> D = COMMON DIFFERENCE BETWEEN TWO CONSECUTIVE TERMS

=> S = SUM

=> S N = SUM OF N TERMS

=> N = NO. OF TERM

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GIVEN ;

IN AN AP :

=> FIRST TERM :

=> T 1 = 2

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SUM OF FIRST 5 TERMS :

=> 1 / 4 OF THE SUM OF NEXT 5 TERMS

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FOR SOLVING :

=> STEP 1 :

=> TO SOLVE THIS PRO. WE NEED T 1 , T5 , T6 , T 10

=> SO TO FIND T 1 , T 5 , T 6 , T 10

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FORMULA FOR TN :

=> TN = A + D ( N - 1 )

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=> T 1 IS GIVEN = 2

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T 5 :

=> T 5 = 2 + D ( 5 - 1 )

=> T 5 = 2 + 4D

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T 6 :

=> T 6 = 2 + D ( 6 - 1 )

=> T 6 = 2 + 5D

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T 10 :

=> T 10 = 2 + D ( 10 - 1 )

=> T 10 = 2 + 9D

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STEP 2

=> ACCORDING TO CONDITION FIND THE SUMS

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HERE IS THE FORMULA FOR SN

=> S N = N / 2 [ A + T N ]

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FROM THE FORMULA ,

=> S5 ( SUM OF 5 TERMS ) = 5 / 2 ( 2 + 2 + 4D )

=> S 5 = 5 / 2 ( 4 + 4D )

=> S 5 = 5 * ( 2 + 2D )

=> S 5 = 10 + 10D

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SUM OF NEXT 5 TERMS :

=> S 5 = 5 / 2 ( 2 + 5D + 2 + 9D )

=> S 5 = 5 / 2 ( 4 + 14 D )

=> S 5 = 5 ( 2 + 7D )

=> S 5 = 10 + 35D

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STEP 3 :

=> ACCORDING TO CONDITION ,

=> SUM OF 1ST 5 TERMS IS 1/4TH OF NEXT FIVE TERMS

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10 + 10D = 1 / 4 * ( 10 + 35 D )

=> 4 ( 10 + 10D ) = 10 + 35D

=> 40 + 40D = 10 + 35D

=> 40 - 10 = 35D - 40D

=> 30 = -5D

=> D = 30 / -5

=> D = -6

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STEP 4 :

=> TO FIND :

=> THE SUM OF FIRST 20 TERMS :

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T 20 :

=> T20 = A + D ( 20 - 1 )

=> T 20 = 2 + ( -6 ) ( 19 )

=> T 20 = 2 - 114

=> T 20 = -112

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S 20 :

=> S 20 = N / 2 [ T 1 + T 20 ]

=> S 20 = 20 / 2 ( 2 + ( - 112 )

=> S 20 = 10 ( - 110 )

=> S 20 = - 1100

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=> THANKS .....

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