Question

Q10 -

If N , the set of natural numbers is partitioned into groups S₁= {1} , S₂= { 2 , 3 } , S₃= { 4 ,5,6,7} , S₄= { 8,9,10,11,12,13,14,15} , ... then find the sum of the number in S₅₀ .

Answer 1

★ PROGRESSION AT IT'S BEST ★

USING MATHEMATICAL GENERALITY AT IT'S BEST

THE NUMBER OF TERMS IN THE GROUP ARE AS FOLLOWS : 1 , 2 , 2² , It's cube and so on ...

The Number of terms in the 50th group =

2⁵⁰⁻¹ = 2⁴⁹

Hence , predicting the consequences of sequences generated :

The 1st term of 1st group = 2⁰ = 1 = 2¹⁻¹

The 1st term of 2nd group = 2¹ = 2²⁻¹ = 2

... and so on

So , the first term of the 50th group will be
the above shown result

Using the standard formula for obtaining sums of ARITHMETIC PROGRESSION -
S( n ) = n/2 [ 2a + ( n - 1 ) d ]

S₅₀ = 2⁴⁹ /2 { 2 × 2⁴⁹ + ( 2⁴⁹ - 1 ) × 1 } =

2⁴⁸ [ 2⁵⁰ + 2⁴⁹ - 1 ]

→→→

2⁴⁸ ( 3 × 2⁴⁹ - 1 )

★✩★✩★✩★✩★✩★✩★✩★✩★✩★✩★

Answer 2

Hey mate ,

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