Question

If the sum of the first 40 terms of the series, 3 + 4 + 8 + 9 + 13 + 14 + 18 + 19 + ….is (102)m, then m is equal to:
(a) 20
(b) 10
(c) 5
(d) 25

Answer 1

Hint: First of all, consider the series given in the question and combine two – two terms from beginning to make it an arithmetic series with a constant common difference. Now, find the sum of the terms of A.P by using \[{{S}_{n}}=\dfrac{n}{2}\left[ 2a+\left( n-1 \right)d \right]\]. Finally, equate this sum with 102m to find the value of m.
Complete step-by-step answer:
We are given that the sum of the first 40 terms of the series, 3 + 4 + 8 + 9 + 13 + 14 + 18 + 19 + …. is (102)m. We have to find the value of m. Let us consider the series given in the question.
S = 3 + 4 + 8 + 9 + 13 + 14 + 18 + 19…… 40 terms
By combining two – two terms from the beginning of the above series, we get,
S = (3 + 4) + (8 + 9) + (13 + 14) + (18 + 19)….. 40 terms
By adding the terms of each bracket in the above series, we get,
S = (7) + (17) + (27) + (37)….. 20 terms
As we have combined two – two terms of the given series, so the above series would have 20 terms. Now, we can see that each term of the above series differs by 10 from its previous term. So, the above series is in AP or arithmetic progression.
We know that the sum of n terms of A.P is given by \[{{S}_{n}}=\dfrac{n}{2}\left[ 2a+\left( n-1 \right)d \right]\] where n = number of terms, a = first term, d = common difference. Let us consider the AP we have got
S = 7 + (17) + (27) + (37) ….. 20 terms
In the above series, we can see that, n = 20 terms, a = 7 and d = 10
So, we get,
\[S=\dfrac{20}{2}\left[ 2.\left( 7 \right)+\left( 20-1 \right)10 \right]\]
\[S=10\left[ 14+19\times 10 \right]\]
\[S=10\left[ 14+190 \right]\]
\[S=2040\]
We are given that the sum is equal to 102m. So, we get,
2040 = 102m
\[m=\dfrac{2040}{102}\]
m = 20
So, we get the value of m as 20.
Hence, option (a) is the right answer.

Note: In this question, many students combine the alternate terms and form two APs as follows: (3 + 8 + 13 + 18…. 20 terms) + (4 + 9 + 14 + 19….. 20 terms). Then they find the sum of two APs individually and add them to get the total sum. This method is also correct but is lengthy as compared to what we have done. So, students should properly analyze the series and solve it accordingly to get the correct answer and also does not lead to calculation mistakes. Also, in this question, many students make this mistake of taking the number of terms as 40 which is finding the sum which is wrong because we have to combine each pair of terms into 1. So, the number of terms will reduce to half that is 20 from 40.