Question

Sum of `n` terms of a certain series is given by `S_n=2n+3n^2`, what is the type of the series and what is its `20^(th)` term?

Answer 1

It is an arithmetic progression with first term as `5` and common difference as `6` and `20^(th)` term is `119`

Explanation:

As sum of `n` terms of a certain series is given by `S_n=2n+3n^2`,

Sum of `20` terms is `2×20+3×20^2=40+1200=1240`.

Further, sum of `19` terms is `2×19+3×19^2=38+1083=1121`,.
Hence `20^(th)` term is `1240-1121=119`.

As sum of `1` term is `2×1+3×1^2=5`, sum of first two terms is `2×2+3×2^2=4+12=16`, second term is `16-5=11` and common difference is `11-5=6`. If it is arithmetic progression the third term should be `11+6=17`.

As sum of first three terms is `2×3+3×3^2=6+27=33`, third term is `33-16=17` hence it is an arithmetic progression.

Answer 2

`d=6,a_20=119,S_20=1240`

Explanation:

`"calculate the first 'few' terms of the sequence"`

`"using "S_n=2n+3n^2`

`S_1=2+3=5rArra_1=5`

`S_2=4+12=16`

`rArra_2=S_2-S_1=16-5=11`

`S_3=6+27=33`

`rArra_3=S_3-S_2=33-16=17`

`S_4=8+48=56`

`rArra_4=S_4-S_3=56-33=23`

`"the first 4 terms are "5,11,17,23`

`"common difference ( d)"`

`d=23-17=17-11=11-5=6`

`rArr" these terms are an arithmetic sequence with "d=6`

`"the sum to n terms of an arithmetic sequence is"`

`•color(white)(x)a_n=a_1+(n-1)d`

`rArra_20=5+(19xx6)=119`

`rArrS_20=(2xx20)+(3xx20^2)=1240`