In an arthmetic sequence of positive numbers the sommon difference is twice the first term and he sum of the first six terms is equal to the square of the first term. Find the first term in the sequence.

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- Mar 28th 2007, 04:26 PMDragonArithmetic Sequences
In an arthmetic sequence of positive numbers the sommon difference is twice the first term and he sum of the first six terms is equal to the square of the first term. Find the first term in the sequence.

- Mar 28th 2007, 06:16 PMJhevon
Note that the terms of an arithmetic series is given by the formula:

a_n = a_1 + (n – 1)d

where a_n is the current term, a_1 is the first term, d is the common difference.

Also note, that the sum of an arithmetic sequence is given by:

S_n = n(a_1 + a_n)/2 = n[2a_1 + (n – 1)d]/2

Now we have the common difference d = 2a_1, and the sum of the first 6 terms, S_6 = 6(a_1 + a_6)/2 = a_1^2

=> a_n = a_1 + (n – 1)2a_1 = a_1 + 2na_1 – 2a_1 = (2n – 1)a_1

=> a_6 = 11a_1

Now S_6 = a_1^2

=> 6(a_1 + a_6)/2 = a_1^2

=> 2a_1^2 = 6a_1 + 6a_6

=> 2a_1^2 = 6a_1 + 6(11a_1)

=> 2a_1^2 = 6a_1 + 66a_1

=> 2a_1^2 = 72a_1 …………………this is a quadratic equation in a_1, you can replace a_1 with a variable, say x, if all the symbols are getting you confused.

=> 2a_1^2 – 72a_1 = 0

=>2a_1(a_1 – 36) = 0

Since a_1 cannot be zero,

a_1 – 36 = 0

=> a_1 = 36

So the first term of the sequence is 36, and if you're interested, the sequence is**a_n = 36 + (n – 1)72**

That is 36, 108, 180, 252, 324, 396,…