If a 1 + a 2 + a 3 + … + a n = 3 2 n + 1 - 2 , for every n ≥ 1 , then a 11 equals

We have given here this mathematical expression which Is known as equation:- If a 1 + a 2 + a 3 + … + a n = 3 2 n + 1 - 2 -------------( 1 ), for every n ≥ 1 Now by using this we just need to find the term a 11 equals:But we also know the formula for the n t h term of an A P Is, T n = a + ( n - 1 ) d , a , d ∈ Z , I & We also know the formula to find the sum of n terms in A P Is S n = n 2 2 a + ( n - 1 ) d , in which a = first term, n = number of terms, and d = common difference between consecutive terms. In equation ( 1 ) put n = 11 then it will become as like expression: ( a 1 + a 2 + a 3 + a 4 + a 5 + a 6 + a 7 + - - - + a 11 ) = 3 ( 2 11 + 1 - 2 ) ( a 1 + a 2 + a 3 + a 4 + a 5 + a 6 + a 7 + - - - + a 11 ) = 3 ( 2 12 - 2 ) ------------( 2 ) As the same when we substitute n = 10 then: ( a 1 + a 2 + a 3 + a 4 + a 5 + a 6 + a 7 + - - - + a 10 ) = 3 ( 2 10 + 1 - 2 ) ( a 1 + a 2 + a 3 + a 4 + a 5 + a 6 + a 7 + - - - + a 10 ) = 3 ( 2 11 - 2 ) --------------( 3 ) Now subtract from equation ( 2 ) to ( 3 ) - ( a 1 + a 2 + a 3 + a 4 + a 5 + a 6 + a 7 + - - - + a 11 ) - ( a 1 + a 2 + a 3 + a 4 + a 5 + a 6 + a 7 + - - - + a 10 ) = 3 ( 2 12 - 2 ) - 3 ( 2 11 - 2 ) a 11 = 3 ( 2 12 - 2 11 ) a 11 = 3 × 2048 = 6144

If ( 2 n + 1 ) + ( 2 n + 3 ) + ( 2 n + 5 ) + … + ( 2 n + 47 ) = 5280 , then what is the value of 1 + 2 + 3 + … + n ?

4851 ( 2 n + 1 ) + ( 2 n + 3 ) + ( 2 n + 5 ) + . . . + ( 2 n + 47 ) = 5280 . ( 2 n + 5 ) - ( 2 n + 3 ) = 2 ( 2 n + 3 ) - ( 2 n + 1 ) = 2 . On the Left-hand Side, ( 2 n + 1 ) + ( 2 n + 3 ) + ( 2 n + 5 ) + . . . . . . + ( 2 n + 47 ) Is an arithmetic progression with common difference 2 . First term of the arithmetic progression is ( 2 n + 1 ) And the last term is ( 2 n + 47 ) . Let the number of terms be T ( 2 n + 47 ) = ( 2 n + 1 ) + ( T - 1 ) 2 . 2 T - 2 = 2 n + 47 - 2 n - 1 2 T = 46 + 2 ⇒ 2 T = 48 ⇒ T = 24 Number of terms of the arithmetic progression = 24 Sum of arithmetic progression = 24 2 × [ ( 2 n + 1 ) + ( 2 n + 1 ) + ( 24 - 1 ) 2 ] 2 T - 2 = 2 n + 47 - 2 n - 1 2 T = 46 + 2 ⇒ 2 T = 48 ⇒ T = 24 Number of terms of the arithmetic progression = 24 Sum of arithmetic progression = 24 2 × [ ( 2 n + 1 ) + ( 2 n + 1 ) + ( 24 - 1 ) 2 ] ⇒ 24 2 [ ( 2 n + 1 ) + ( 2 n + 1 ) + ( 24 - 1 ) 2 ] = 5280 = 24 2 [ 4 n + 2 + 23 x 2 ] = 5280 ⇒ 24 2 [ 4 n + 48 ] = 5280 ⇒ 48 [ n + 12 ] = 5280 ⇒ n + 12 = 110 ⇒ n = 110 - 12 ⇒ n = 98 The value of 1 + 2 + 3 + . . . + n = 1 + 2 + 3 + . . . . . + 98 ⇒ 98 × ( 98 + 1 ) 2 ⇒ 49 x 99 ⇒ 4851 The value of 1 + 2 + 3 + . . . . + n Is 4851 . Hence, the correct answer is 4851 . .