Consider the sum sequence (an)n≥0 = 4, 11, 18, 25, 32, ....(a) What type of sequence is (an)n≥o? Find an explicit formula an.(b) Is 249 in the sequence? Explain.(c) How many summands are in the sum 4 + 11 + 18 + 25 + . . . + 249?(d) Compute the sum using Gauss' trick.

Answered: Consider the sum sequence (an)n>0 = 4,…

Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

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Consider the sum sequence (an)n≥0 = 4, 11, 18, 25, 32, ....
(a) What type of sequence is (an)n≥o? Find an explicit formula an.
(b) Is 249 in the sequence? Explain.
(c) How many summands are in the sum 4 + 11 + 18 + 25 + . . . + 249?
(d) Compute the sum using Gauss' trick.

Expert Solution

Step 1: Solution

(a) Given ${(a_{n})}_{n \geq 0} = 4, 11, 18, 25, 32, . . . .$

$a_{1} = 4 a_{2} = a_{1} + (7) a_{3} = a_{2} + 7 = a_{1} + 2.7 a_{4} = a_{3} + 7 = a_{1} + 3.7$

Therefore we can notice that the common difference $7$ which we can compute by $a_{n} - a_{n - 1}$

The first term is $a_{1} = 4$

Therefore the $n^{t h}$ term is

$a_{n} = a + d (n - 1) = 4 + 7 (n - 1) \Rightarrow a_{n} = 7 n - 3$

Hence $a_{n} = 7 n - 3$

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Consider the sum sequence (an)n>0 = 4, 11, 18, 25, 32, … (a) What type of sequence is (an)nzo? Find an explicit formula an. (b) Is 249 in the sequence? Explain. (c) How many summands are in the sum 4 + 11 + 18+ 25 + (d) Compute the sum using Gauss' trick. • +249?