Question

we are given a sequence of five non-zero numbers where the sum of each term and its neighboring terms is 15 or 25. Find the sum of all the terms in the sequence.

Answer 1

Let the series be a,a+d,a+2d,...,a+(n−1)d,a+(3n−1)d,a+3nd.

Also,given that the sum of first 3n terms is equal the sum of next n terms.

⇒

2

3n

[2a+(3n−1)d]=

2

n

[2(a+3nd)+(n−1)d]

⇒3[2a+3nd−d]=[2a+6nd+nd−d]

⇒6a+9nd−3d=2a+7nd−d

⇒6a+9nd−3d−2a−7nd+d=0

⇒4a+2nd−2d=0

⇒2(2a+(n−1)d)=0

⇒2a+(n−1)d=0 ...(1)

Now,

S

2n(next2nterms)

S

2n

=

2

2n

[2(a+2nd)+(2n−1)d]

2

2n

[2a+(2n−1)d]

=

2a+4nd+2nd−d

2a+(2n−1)d

=

2a+nd+5nd−d

2a+nd+nd−d

=

2a+(n−1)d+5nd

2a+(n−1)d+nd

=

5nd

nd

...(from(1))

=

5

1

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