Question

D=3, a10=27.5 Sn=? a1= ?, n=?

arithmethic Series

Answer 1

`a_1 = a = 0.5`

`S_10 = 140`

Explanation:

Arithmetic Sequence , `a_n = a + (n-1) d`

`a_n` is `n^(th)` term, `a` the first term and `d` the common difference.

Given `a_n = 27.5, d = 3`, to find sum of first n terms `S_n`

`S_n = n/2 (2a + (n-1) d`

`a + (10-1)*3 = 27.5`

`a = 27.5 - (10-1)*3 = 27.5 - 27 = 0.5`

Sum of first 10 terms, `S_10 = (10/2) * (2*0.5 + (10-1)*3)`

`S_10 = 5 * (1 + 9*3) = 140`

Verification:

`S_n = n/2 (a + l)` where l is the last term.

`S_10 = (10/2) * (a + a_10)`

`S_10 = 5 * (0.5 + 27.5) = 140`

Answer 2

`" "`

First term : `color(red)(a_1=0.5`

Last term : `color(red)(a_100=297.50`

Total number of terms : `color(red)(n=100`

Sum to "n" terms : `color(red)( S_100=14,900`

Explanation:

`" "`
Given :

Common difference : `color(red)(d=3`

Tenth term of the arithmetic sequence: `color(red)(a_10=27.5`

`color(red)("What is expected ?"`

Sum to "n" terms : `color(blue)(S_n`

First term : `color(blue)(a_1`

Total number of terms in the arithmetic series : `color(blue)(n`

Using `color(red)(d and a_10`, we can find the first term, which is indicated by `color(red)(a_1`

Formula used to find a specific term:

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

We will use this formula to find the first term : `color(red)(a_1`

Substitute all known values in the formula:

`color(blue)(27.5=a_1+(10-1)3`

`rArr 27.5=a_1+(9)(3)`

`rArr 27.5=a_1+27`

Subtract `color(blue)(27` from both sides of the equation to isolate `color(blue)(a_1`

`rArr 27.5-27=a_1+27-27`

`rArr 0.5=a_1+cancel 27-cancel 27`

`rArr color(red)(a_1=0.5`

Hence, we have our first term of the arithmetic series: `color(red)(a_1=0.5`

We now have the following useful details:

First term : `color(red)(a_1=0.5`

Tenth term: `color(red)(a_10=27.5`

Common difference : `color(red)(d=3`

The formula to find the Sum of "n" terms in any arithmetic series is given by:

`color(red)(S_n=[n(a_1+a_n)]/2`

OR

`color(red)(S_n=[(n/2)(a_1+a_n)]`

We can find the SUM to 20, 50, 100 or more terms of the arithmetic series using the above formula:

Assume that we want to find the SUM to 100 terms:

i.e., find `color(red)(S_100`

We will use the following formula to find a specific term:

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

We know that `a_1=0.5; d = 3; n=100`

We can find `color(red)(a_100`

`rArr a_100 =a_1+(n-1)3`

`rArr a_100=0.5+(100-1)3`

`rArr a_100=0.5+(99)(3)`

`rArr a_100=0.5+297`

`rArr a_100=297.50`

To find the sum to 100 terms use the formula:

`color(red)(S_n=[(n/2)(a_1+a_n)]`

`rArr S_100=(100/2)(0.5+297.50)`

`rArr S_100=(50)(298)`

Hence, `color(red)(S_100 = 14,900`

Hope you find this solution useful.