# In an arithmetic series, t20=93 and s25=1625 find the first term a and the common difference d ?(Hint: systems of equations might be helpful here)

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### 2 Answers

An arithmetic sequence can be given as below.

`t_n = t_1 + (n-1) + d`

`s_n = n/2 [2t_1 + (n-1)d]`

a_1 is the first term and d is the common difference.

`t_20 = t_1 + (20-1)d`

`93 = t_1 + 19d`

`s_25 = 25/2[2t_1 + (24-1)d]`

`130 = [2t_1 + 24d]`

`130 = 2t_1 + 24d`

Solving these two equations,

`56 = 14d`

`d = 4`

**The common difference is 4.**

Substituting in first equation,

`93 = t_1 + 19xx4`

`93 = t_1 + 76`

`t_1 = 17`

**The first term is 17.**

**Sources:**

### User Comments

As seen in the others guys explanation, the common deference is 4 yes. And yes the first term is 17. I am just writting this to let you know that hhe is correct and i have done this ,myself so you do not have to worry about this answer possibly being incorrect because he is ABSOLUTELY CORRECT!! so as you have probably realized at this point in your life, that math is the center of the universe without math nothing could ever come to be, and so this is just me telling you to treasure math and learn to love it becasue without math you would not be here right now and i would not be giving you this explanation! so have a good day, and life. and Always know that math is the most important subject in school.