Show that the progressions given below is an AP. Find the first term, common difference and next term of each given series.
(1) 9, 15, 21, 27, ….
(ii)11, 6, 1, -4…
(iii)`-1, (-5)/(6), (-2)/(3), (-1)/(2),….`
(iv)`sqrt(2), sqrt(8), sqrt(18), sqrt(32), ....`
`(v)sqrt(20), sqrt(45), sqrt(80), sqrt(125),...`

Answer

`(i) a=9, d = 6, T_(5) = 33` (ii) a = 11, d = -5, `T_(5)` = -9
(iii) `a = -1, d = (1)/(6), T_(5) = (-1)/(3)` (iv) `a = sqrt(2),d = sqrt(2), T_(5) = sqrt(50)`
`(v) a = sqrt(20), d = sqrt(5), T_(5) = sqrt(180)`

Explanation

(iv) Given progression is `sqrt(2), sqrt(4 xx 2), sqrt(9 xx 2), sqrt(16 xx 2),..`
i.e., `sqrt(2), 2sqrt(2), 3sqrt(2), 4sqrt(2),…`
`T_(5) = 5sqrt(2) = sqrt(5 xx 5 xx 2) = sqrt(50)`
(v) Given progression is `sqrt(4 xx5), sqrt(9 xx 5), sqrt(16 xx 5), sqrt(25 xx 5),...`
i.e., `2sqrt(5), 3sqrt(5), 4sqrt(5), 5sqrt(5)`,...
`T_(5) = 6sqrt(5) = sqrt(6 xx 6 xx 5) = sqrt(180).`