Question

Can the sum of the first few consecutive terms of the arithmetic sequence 5, 7, 9, ….be 140? What about 240?

Answer 1

Let the first term of the A.P. be a + b and the common difference be a.

The given sequence is 5, 7, 9, …

∴ a = 7 − 5 = 2

a + b = 5

⇒ 2 + b = 5

⇒ b = 5 − 2 = 3

The general term of an A.P. is given by

∴

Let the sum of n terms of the given sequence be 140.

Sum of n terms =

Comparing the above equation with general quadratic equation:

a = 2, b = 8 and c = −280

Discriminant of the above equation is:

b² − 4ac = (8)² − 4(2)(−280)

= 64 + 2240

= 2304

As the discriminant is positive, the above equation has two solutions.

As n is the number of terms, n cannot be negative.

∴ n = 10

Therefore, the sum of 10 consecutive terms of the given sequence is 140.

Now, let the sum of m terms of the given sequence be 240.

Sum of m terms =

Comparing the above equation with general quadratic equation:

a = 2, b = 8 and c = −480

Discriminant of the above equation is:

b² − 4ac = (8)² − 4(2)(−480)

= 64 + 3840

= 3904

As the discriminant is positive, the above equation has two solutions.

As n is the number of terms, n cannot be negative or in decimals.

Therefore, 240 cannot be the sum of the first few consecutive terms of the given sequence.