Arithmetic Sequence

five consecutive terms of an arithmetic sequence have a sum of 40. The product of the first, middle and the last term is 224. Find the terms of sequence PLEASE HELP !

I.

\(\begin{array}{|rcll|} \hline a_i + a_{i+1}+ a_{i+2}+ a_{i+3}+ a_{i+4} &=& 40 \\ a_i + (a_i+d) + (a_i+2d)+ (a_i+3d)+ (a_i+4d) &=& 40 \\ 5a_i + 10d &=& 40 \qquad & | \qquad :5\\ a_i + 2d &=& 8 \\ \mathbf{a_i} & \mathbf{=} & \mathbf{ 8 - 2d }\\ \hline \end{array}\)

II.

\(\begin{array}{|rcll|} \hline a_i \cdot a_{i+2} \cdot a_{i+4} &=& 224 \\ a_i \cdot (a_i+2d) \cdot (a_i+4d) &=& 224 \qquad & | \qquad a_i = 8 - 2d \\ (8 - 2d) \cdot (8 - 2d+2d) \cdot (8 - 2d+4d) &=& 224 \\ (8 - 2d) \cdot 8 \cdot (8 + 2d) &=& 224 \qquad & | \qquad :8 \\ (8 - 2d) \cdot (8 + 2d) &=& \frac{224}{8} \\ (8 - 2d) \cdot (8 + 2d) &=& 28 \qquad & | \qquad (u-v)\cdot (u+v)=u^2-v^2 \\ 8^2-(2d)^2 &=& 28 \\ 64-4d^2 &=& 28 \\ 4d^2 &=& 64-28 \\ 4d^2 &=& 36 \qquad & | \qquad :4 \\ d^2 &=& 9 \\ d &=& \sqrt{9} \\ \mathbf{d} & \mathbf{=} & \mathbf{ 3 }\\ \hline \end{array}\)

III.

\(\begin{array}{|rcll|} \hline a_i &=& 8-2d \qquad & | \qquad d = 3 \\ a_i &=& 8-2\cdot 3 \\ a_i &=& 8-6 \\ \mathbf{a_i} & \mathbf{=} & \mathbf{ 2 }\\ \hline \end{array}\)

five consecutive terms of an arithmetic sequence: \(\dots, 2, 5, 8, 11, 14, \dots\)

\(\begin{array}{|lrcll|} \hline \text{sum: } & 2+5+8+11+14 = 40 \\ \text{product: } & 2\cdot 8 \cdot 14 = 224 \\ \hline \end{array}\)

The Series is: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32......etc. The common difference =3

2 x 8 x 14=224

2+5+8+11+14=40

Let the terms be   a, a + d, a + 2d, a + 3d, a + 4d   = 40....simplify

5a + 10d = 40     divide through by 5

a + 2d  = 8    →  a = 8 - 2d

And we know that

a (a + 2d)(a + 4d)  = 224   substitute for a

(8 - 2d)(8 - 2d + 2d)(a - 2d + 4d)  = 224    simplify

(8 - 2d)(8)(8 + 2d)  = 224     divide through by 8

(8 - 2d)(8 + 2d)  = 28   simplify

64 - 4d^2  = 28

4d^2 - 36 = 0

d^2 - 9 = 0   factor

(d + 3) (d - 3)  = 0    do d = 3  or d = -3      reject - 3

So.....a =  8 - 2d →   8 - 2(3)  = 2

And the terms are

2 + 5 + 8 + 11 + 14 =  40

Thank you all !!
you were really helpfull