Manipal Entrance Test 2013 Math Paper

The sequence of differences between successive terms is 2, 6,18, 54......
Clearly, it is a GP. Let Tn be the n ‌th term of the given series and Sn be the sum of its n terms.
Then,
Sn=5+7+13+31+85+....+Tn−1+Tn ....(i)
Sn=5+7+13+31+....+Tn−1+Tn ......(ii)
Subtracting (ii) from (i), we get
O=5+[2+6+18+54+.....+(Tn−Tn−1)]−Tn
⇒ O=5+2.

3n−1−1

3−1

−Tn
⇒ Tn=5+(3n−1−1)=4+3n−1
∴ Sn=

n

∑

k=1

Tk=

n

∑

k=1

(4+3k−1)
=

n

∑

k=1

4+

n

∑

k=1

3k−1
=4n+(1+3+32+....+3n−1)
=4n+1×(

3n−1

3−1

)
=4n+(

3n−1

2

)
=

1

2

(3n+8n−1)