Question

How do you find the first three terms of the arithmetic series `a_1=17`, `a_n=197`, and `S_n=2247`?

Answer 1

First three terms are `17,26 and 35`

Explanation:

`1`st term is `a_1=17 , a_n=197` , common difference is `b`

And number of terms is `n`.

Mid term is `a_m=(17+197)/2=107`

Sum `S_n+= a_m*n or 2247 = 107 * n :. n= 2247/107=21`

`a_n= a_1+(n-1)b or 197 = 17+ (21-1)b or 20b=180` or

`b=180/20=9 :. a_2=a_1+b=17+9=26`

`a_3=a_1+2b=17+2*9=17+18=35`

First three terms are `17,26 and 35` [Ans]

Answer 2

`17,26,35...`

Explanation:

An arithmetic sequence is the one in which the difference of the successive terms is the same

For example,

`1,3,5,7..` is an arithmetic sequence because the common difference is `2`. The symbol that i will use for the common difference here is `d`

Now lets solve the problem. For that we use the formula

`color(green)(S_n=n((a_1+a_n)/2)`

Where, `S_n` is the sum of the first `n` terms, `a_1` is the first term and `a_n` is the `n^(th)` term. We already knew the values, so apply them

`rarr2247=n((197+17)/2)`

`rarr2247=n(214)`

`color(green)(rArrn=21`

Now we know that the `21^(st)` term is `197`

To find the common difference, we use

`color(green)(a_n=a_1+(n-1)d`

`rarr197=17+(21-1)d`

`rarr180=20d`

`color(green)(rArrd=9`

Now we have come to an end to the answer. We add `9` to each of the successive terms and we get

`color(purple)(17,26,35....`

Hope this helps!!! ☺