Find the 3 consecutive terms in ap whose sum is 18 and sum of their sq

1.	Find the 3 consecutive terms in ap whose sum is 18 and sum of their squares is 140
Answer» For any AP, if n is odd, $S_{n} = n * [\frac{n + 1}{2}]^{th} term$ ( $S_{n}$ means sum of any n consecutive terms of the sequence, not first n consecutive terms.) Here, $S_{3} = 3 * [\frac{3 + 1}{2}]^{th} term \\ \\ = 3 * T_{2} = 18 \\ \\ T_{2} = \frac{18}{3} = 6$ ∴ The MIDDLE term of the 3 consecutive terms is 6. 6² = 36. 140 - 36 = 104 is the sum of the squares of the first and the third terms. Let the first and the third terms be (6 - d) and (6 + d) respectively. (d = COMMON difference) $(6 - d)^2 + (6 + d)^2 = 104 \\ \\ = 36 - 12d + d^2 + 36 + 12d + d^2 = 104 \\ \\ = 72 + 2d^2 = 104 \\ \\ 2d^2 = 104 - 72 = 32 \\ \\ d^2 = \frac{32}{2} = 16 \\ \\ d = \sqrt{16} = <klux>4</klux>$ ∴ 4 is the common difference. $T_{1} = 6 - 4 = 2 \\ \\ T_{3} = 6 + 4 = 10$ ∴ The NUMBERS are 2, 6, and 10. Hope this may be helpful. Please mark my answer as the brainliest if this may be helpful. Thank you. Have a nice day.

Find the 3 consecutive terms in ap whose sum is 18 and sum of their squares is 140

Answer»

For any AP, if n is odd,

$S_{n} = n * [\frac{n + 1}{2}]^{th} term$

( $S_{n}$ means sum of any n consecutive terms of the sequence, not first n consecutive terms.)

Here,

$S_{3} = 3 * [\frac{3 + 1}{2}]^{th} term \\ \\ = 3 * T_{2} = 18 \\ \\ T_{2} = \frac{18}{3} = 6$

∴ The MIDDLE term of the 3 consecutive terms is 6.

6² = 36.

140 - 36 = 104 is the sum of the squares of the first and the third terms.

Let the first and the third terms be (6 - d) and (6 + d) respectively. (d = COMMON difference)

$(6 - d)^2 + (6 + d)^2 = 104 \\ \\ = 36 - 12d + d^2 + 36 + 12d + d^2 = 104 \\ \\ = 72 + 2d^2 = 104 \\ \\ 2d^2 = 104 - 72 = 32 \\ \\ d^2 = \frac{32}{2} = 16 \\ \\ d = \sqrt{16} = <klux>4</klux>$

∴ 4 is the common difference.

$T_{1} = 6 - 4 = 2 \\ \\ T_{3} = 6 + 4 = 10$

∴ The NUMBERS are 2, 6, and 10.

Hope this may be helpful.

Please mark my answer as the brainliest if this may be helpful.

Thank you. Have a nice day.

Find the 3 consecutive terms in ap whose sum is 18 and sum of their squares is 140

Discussion

No Comment Found

Related InterviewSolutions

Reply to Comment