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11 Jan 2022, 03:30
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D
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Difficulty:
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59% (03:18) correct
41%
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wrong
based on 125
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The sum of the squares of the first and fifth term of an arithmetic progression is 218. If the third term is less than the first term, and the sum of these two terms is 16, what is the value of the first term?
A. 10
B. 11
C. 12
D. 13
E. 14
A. 10
B. 11
C. 12
D. 13
E. 14
yashikaaggarwal
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18 Jan 2022, 21:22
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The sum of the squares of the first and fifth term of an arithmetic progression is 218.
A^2+(a+4d)^2 = 218
2a^2+ 16d^2+ 8ad = 218
A^2+8d^2+4ad = 109 (1)
If the third term is less than the first term, and the sum of these two terms is 16.
A+A+2D = 16
A+D = 8 -------(2)
Squaring second equation.
A^2+D^2+2AD = 64
Using elimination,
A^2+8d^2+4ad = 109
-A^2-D^2-2AD = -64
--------------------
7D^2+2AD = 45
D(7D+2A) = 45
D(5D+16) = 45 (from eq 2)
5D^2 + 16D - 45 = 0
5D^2 + 25D - 9D - 45 = 0
5D(D+5) -9 (D+5) = 0
D = -5 and 9/5
We know D is negative. As 3rd term of series is less than first term.
Therefore, D = -5
A+D = 8
A-5 = 8
A = 13
Answer is D option
Posted from my mobile device
A^2+(a+4d)^2 = 218
2a^2+ 16d^2+ 8ad = 218
A^2+8d^2+4ad = 109 (1)
If the third term is less than the first term, and the sum of these two terms is 16.
A+A+2D = 16
A+D = 8 -------(2)
Squaring second equation.
A^2+D^2+2AD = 64
Using elimination,
A^2+8d^2+4ad = 109
-A^2-D^2-2AD = -64
--------------------
7D^2+2AD = 45
D(7D+2A) = 45
D(5D+16) = 45 (from eq 2)
5D^2 + 16D - 45 = 0
5D^2 + 25D - 9D - 45 = 0
5D(D+5) -9 (D+5) = 0
D = -5 and 9/5
We know D is negative. As 3rd term of series is less than first term.
Therefore, D = -5
A+D = 8
A-5 = 8
A = 13
Answer is D option
Posted from my mobile device
19 Jan 2022, 05:26
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Bunuel
We know that the series is decreasing
Let the series be a, a-b, a-2b, a-3b, a-4b ...
Where \(a\) is the first term and \(b\) is the common difference
\((a ) + (a-2b )= 16\)
\(2a-2b=16\)
\(a-b=8 \)
So the second term is \(8 \)
Then we can use the options
If the first term is \(14\) and second term is \(8 \) then \(b = -6\), then \(5^{th}\) term \(= -10\)
\((14)^2 + (-10)^2 = 296 \neq 218\) Eliminate this option
If the first term is \(13\) and second term is \(8\) then \(b =-5,\) then \(5^{th}\) term is \(= -7\)
\((13)^2 + (-7)^2 = 218 \) correct
Hence first term must be \(13\)
Ans D
