Method of Differences

Method of Differences

(a) If a₁, a₂, . . ., an is a sequences such that a₂-a₁, a₃-a₂, . . ., is either an AP or GP. Then the nth term of the sequence can be found as follows.

Let S = 5 + 11 + 19 + . . . + t_n-1 + t_n

S = 5 + 11 + . . . + t_n-2 + tn-1 + _tn

Subtracting we get

0 = 5 + 6 + 8 + 10 + 12 + . . . - t_n

⇒ t_n = n₂ + 3_{n + 1} (calculate)

Having got tn we can find the sum easily

i.e. n(n₂ + 6_{n + 8})

(b) Suppose corresponding to the sequence a₁, a_2, . . . ,an there exists a sequence
b₁, b₂, . . ., bn such that ak = b_k – b_k-1, ∀ k. Then the sum can be calculated as follows. As an illustration.

Let us find out the sum

then

Let .

Then t₁ + t₂ + . . . + t_n = b₀ – b_n =