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## SAT Arithmetic Progression : Sum of an Arithmetic Sequence

### Question 2: Sum of First 25 terms of an arithmetic sequence

What is the sum of the first 25 terms of an arithmetic sequence if the sum of its 8th and 18th term is 72?

1. 1800
2. 600
3. 900
4. 936
5. 450
Choice C. The sum of the first 25 terms is 900.

This is an easy to moderate level difficulty SAT Math Question. Try to solve it using the Hint / Formula provided in the first tab. If you need further assistance, step wise explanation and an alternative shortcut is provided in the subsequent tabs.

The sum of the first n terms of an arithmetic sequence Sn = $n $over 2 \\$ $\ \left\lbrack 2a_1 + \lbrack n-1 \rbrack d \right\rbrack \\$ Where a1 is the first term, 'n' is the number of terms and 'd' is the common difference. To find the sum of the first 25 terms of the arithmetic sequence, you need the first term t1 and the common difference 'd'. Data given: Sum of 8th term and 18th term of the sequence is 72. 1. Express the 8th term and the 18th term in terms of the first term and common difference of the arithmetic progression. 2. Write down the formula to compute the sum of the first 25 terms by filling the value for 'n'. 3. Notice that what is required to get the answer is step 2 is available from the expression in step 1. ### Step 1: Express the sum of the 8th and the 18th term in terms of t1 and d The 8th term of the arithmetic progression, t8 = t1 +$8 - 1)d

or t8 = t1 + 7d

The 18th term of the sequence t18 = t1 + (18 - 1)d

or t18 t1 + 17d

Sum of the 8th and 18th term is 72

t8 + t18 = t1 + 7d + t1 + 17d = 72

or 2(t1) + 24d = 72 .... (eqn 1)

### Step 2: Wthe formula to compute sum of first 25 terms in terms of t1 and d

The sum of the first n terms of an arithmetic sequence Sn = $n $over 2 \\$ $\ \left\lbrack 2t_1 + \lbrack n-1 \rbrack d \right\rbrack \\$ The sum of the first 25 terms of an arithmetic sequence S25 = $25 \over 2 \\$ $\ \left\lbrack 2t_1 + \lbrack 25-1 \rbrack d \right\rbrack \\$ S25 = $25 \over 2 \\$ $\ \left\lbrack 2t_1 + 24d \right\rbrack \\$ ...$eqn 2)

The missing information is the value of $$\left\lbrack 2t_1 + 24d \right\rbrack \\$ In Step 1 in eqn$1), we found the value of $$\left\lbrack 2t_1 + 24d \right\rbrack \\$ as 72 Substituting the value of 2t1 + 24d as 72 in eqn$2), we get S25 = $25 \over 2 \\$ $\ \left\lbrack 72 \right\rbrack \\$

S25 = 25 * 36 = 900

### Sum of an AP is the product of the middle term and the number of terms

Therefore, sum of the first 25 terms S25 = 25 * t13.

We will have the answer when we compute the value of t13

In an AP, the value of any term of the sequence is the arithmetic mean of two terms equidistant from it.

For instance, t2 = $t_1 + t_3 \over 2 \\$ and t5 = $t_2 + t_8 \over 2 \\$

So, t13 = $t_8 + t_{18} \over 2 \\$

We know that t8 + t18 = 72

Hence, t13 = $t_8 + t_{18} \over 2 \\$ = $72 \over 2 \\$ = 36

Sum of the first 25 terms, S25 = 25 * 36

S25 = 900

## Concept Overview : Arithmetic Progression

Get a quick overview on the basic concepts of Arithmetic sequence including formulae to find the nth term of an arithmetic sequence and the sum of n terms of an arithmetic sequence.

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