Sum of the series to n terms 5+7+13+31+85+ is

Correct Answer is : 12[3n+8n−1]1/2[3ⁿ+8n-1]21[3n+8n−1]

Correct Answer is : 12[3n+8n−1]1/2[3ⁿ+8n-1]21[3n+8n−1]

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Manipal Entrance Test 2013 Math Paper

Question : 4 of 80

Marks: +1, -0

5+7+13+31+85+\dots

Solution:

The sequence of differences between successive terms is 2, 6,18, 54......

Clearly, it is a GP. Let

T_n

be the n

^{\text{th}}

term of the given series and

S_n

be the sum of its n terms.

Then,

S_{n}=5+7+13+31+85+\dots +T_{n-1}+T_{n}

....(i)

S_{n}=5+7+13+31+\dots +T_{n-1}+T_{n}

......(ii)

Subtracting (ii) from (i), we get

O=5+ [2+6+18+54+\dots + (T_{n}-T_{n-1}) ]-T_{n}

⇒

O=5+2\cdot\frac{3^{n-1}-1}{3-1}-T_{n}

⇒

T_{n}=5+ (3^{n-1}-1 )=4+3^{n-1}

∴

S_{n}=\sum\limits_{k=1}^{n} T_{k}=\sum\limits_{k=1}^{n} (4+3^{k-1})

=\sum\limits_{k=1}^{n}4+\sum\limits_{k=1}^{n}3^{k}-1

=4n+ (1+3+3^{2}+\dots+3^{n-1})

=4n+1\times\frac{3^{n}-1}{3-1}

=4n+\left(\frac{3^{n}-1}{2}\right)

=\frac{1}{2}(3^{n}+8n-1)

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