Question

How many terms of the AP 63, 60, 57, 54, ... must be taken so that their sum is 693? Explain the double answer

Answer 1

here double answer implies that 22nd term is zero so it doesn't affect the answer and sum will be same

Answer 2

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QUESTION :

How many terms of the AP 63, 60, 57, 54, ... must be taken so that their sum is 693?

GIVEN :

a = 63 , d = a2 - a1 = 60 - 63 = -3 and Sn = 693

SOLUTION :

we know the formula =>

Sn = n/2 [2a +(n-1)×d]

=> 693 = n/2 [2×63+ (n-1)× -3]

=> 693 = n/2 [126 -3n+3]

=> 693 = 126n/2 -3n²/2 + 3n/2

=> 693 = 126n + 3n/2 - 3n²/2

=> 693 = 129n/ 2 - 3n²/2

=> 2×693 = 129n - 3n²

=> 1,386 = 129n - 3n²

=> 1,386 = -3n² + 129n

=> -3n² + 129n - 1,386 = 0

=> - [3n² - 129n + 1,386 ] =0

=> 3n² - 129n + 1386 = 0

=> 3[n² - 43n + 462 ]

=> n²- 43n + 462

=> n² -(22+21)n + 462

=> n² -22n - 21n + 462

=> n(n - 22) -21(n -22 )=0

=> (n-21) (n-22) = 0

=> n = 21, n = 22