The sum of first four terms of an arithmetic sequence is 64 and sum of first 10 terms is 340.what is the sum of first and fourth terms of this sequence?
Answers
\large\underline{\sf{Solution-}}
Solution−
Let assume that
First term of an AP series be a and common difference of an AP series is d.
Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,
↝ Sum of n terms of an arithmetic progression is,
\begin{gathered}\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{S_n\:=\dfrac{n}{2} \bigg(2 \:a\:+\:(n\:-\:1)\:d \bigg)}}}}}} \\ \end{gathered}\end{gathered}
★
S
n
=
2
n
(2a+(n−1)d)
Wʜᴇʀᴇ,
Sₙ is the sum of n terms of AP.
a is the first term of the sequence.
n is the no. of terms.
d is the common difference.
According to statement,
\rm \: S_4 = 64S
4
=64
\rm \: \dfrac{4}{2} \bigg(2a + (4 - 1)d \bigg) = 64
2
4
(2a+(4−1)d)=64
\rm \:2 \bigg(2a + 3d \bigg) = 642(2a+3d)=64
\rm\implies \:2a + 3d = 32 - - - - (1)⟹2a+3d=32−−−−(1)
According to statement again
\rm \: S_{10} = 340S
10
=340
\rm \: \dfrac{10}{2} \bigg(2a + (10 - 1)d\bigg) = 340
2
10
(2a+(10−1)d)=340
\rm \: 5 \bigg(2a + 9d\bigg) = 3405(2a+9d)=340
\rm\implies \:2a + 9d = 68 - - - - (2)⟹2a+9d=68−−−−(2)
On Subtracting equation (1) from equation (2), we get
\rm \: 6d = 366d=36
\rm\implies \:d = 6⟹d=6
On substituting the value of d = 6, in equation (1), we get
\rm \: 2a + 3 \times 6 = 322a+3×6=32
\rm \: 2a + 18 = 322a+18=32
\rm \: 2a = 32 - 182a=32−18
\rm \: 2a = 142a=14
\rm\implies \:a = 7⟹a=7
=a+(n−1)d
Wʜᴇʀᴇ,
aₙ is the nᵗʰ term.
a is the first term of the sequence.
n is the no. of terms.
d is the common difference.
Tʜᴜs,
\rm \: a_1 + a_4a
1
+a
4
\rm \: = \: a \: + \: a \: + \: (4 - 1)d=a+a+(4−1)d
\rm \: = \:2 a \: + \: 3d=2a+3d
\rm \: = \:2 \times 7\: + \: 3 \times 6=2×7+3×6
\rm \: = \: 14 + 18=14+18
\rm \: = \: 32=32
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SHORT CUT TRICK
From equation (1), we have
\rm \: 2a + 3d = 322a+3d=32
can be further rewritten as
\rm \: a + a + 3d = 32a+a+3d=32
\rm\implies \:a_1 + a_4 = 32⟹a
1
+a
4
=32
So, Sum of first and fourth term is 32
Answer:
Hope this ans is helpful for you
Step-by-step explanation:
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