Question

How many terms of the AP : 24, 21, 18, . . . must be taken so that their sum is 78?

Answer 1

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How many terms of the AP : 24, 21, 18, . . . must be taken so that their sum is 78?

Ask for details Follow Report by ChandanMitter3 12.06.2015

Answers

TPS

TPS Genius

First term, a = 24

common difference, d = 21-24 = -3

let the number of terms to get sum 78 is n.

S_n=78\\ \\ \Rightarrow \frac{n}{2}(2a+(n-1)d)=78\\ \\ \Rightarrow \frac{n}{2}(2 \times 24+(n-1)(-3))=78\\ \\ \Rightarrow n(48-3n+3)=78 \times 2\\ \\ \Rightarrow -3n^2+51n-156=0\\ \\ \Rightarrow 3n^2-51n+156=0

Solving the quadratic equation, we get n=4 and n=13.

So you can take either 4 terms or 13 terms to get the sum 78.

Answer 2

ғɪʀsᴛ ᴛᴇʀᴍ, ᴀ = 24

ᴄᴏᴍᴍᴏɴ ᴅɪғғᴇʀᴇɴᴄᴇ, ᴅ = 21-24 = -3

ʟᴇᴛ ᴛʜᴇ ɴᴜᴍʙᴇʀ ᴏғ ᴛᴇʀᴍs ᴛᴏ ɢᴇᴛ sᴜᴍ 78 ɪs ɴ.

sɴ = ɴ/2 (2ᴀ +( ɴ-1) ᴅ)

78 = ɴ/2(48+ (-3)ɴ + 3)

78 = ɴ/2(51 + (-3)ɴ)

78*2=ɴ (51+ (-3)ɴ)

156= 51ɴ + -3ɴ^2

3ɴ^2-51ɴ+156=0

ɴ^2 - 17ɴ +52 =0

ɴ^2-13ɴ-4ɴ +52 =0

ɴ(ɴ-13)-4(ɴ-13)=0

(ɴ-4)(ɴ-13) =0

ɴ = 4,13

sᴏʟᴠɪɴɢ ᴛʜᴇ ϙᴜᴀᴅʀᴀᴛɪᴄ ᴇϙᴜᴀᴛɪᴏɴ, ᴡᴇ ɢᴇᴛ ɴ=4 ᴀɴᴅ ɴ=13.

sᴏ ʏᴏᴜ ᴄᴀɴ ᴛᴀᴋᴇ ᴇɪᴛʜᴇʀ 4 ᴛᴇʀᴍs ᴏʀ 13 ᴛᴇʀᴍs ᴛᴏ ɢᴇᴛ ᴛʜᴇ sᴜᴍ 78.